The end behavior of a polynomial with positive leading coefficient is either "up-up" or "down-down" depending on the degree of the polynomial the end behavior of a polynomial with negative leading coefficient is either "up-down" or "down-up" depending on the degree of the polynomial.
Actually, the end behavior of a polynomial refers to the behavior of the function as the input variable (usually x) approaches positive or negative infinity.
The end behavior of a polynomial is determined by the degree and leading coefficient of the polynomial.
If the degree of the polynomial is even and the leading coefficient is positive, then the end behavior is as follows:
As x approaches negative infinity, the graph of the polynomial approaches the x-axis from above (the arrows point in opposite directions).
As x approaches positive infinity, the graph of the polynomial approaches the x-axis from below (the arrows point in opposite directions).
If the degree of the polynomial is even and the leading coefficient is negative, then the end behavior is as follows:
As x approaches negative infinity, the graph of the polynomial approaches the x-axis from below (the arrows point in opposite directions).
As x approaches positive infinity, the graph of the polynomial approaches the x-axis from above (the arrows point in opposite directions).
If the degree of the polynomial is odd and the leading coefficient is positive, then the end behavior is as follows:
As x approaches negative infinity, the graph of the polynomial goes down to negative infinity (the arrows point in the same direction).
As x approaches positive infinity, the graph of the polynomial goes up to positive infinity (the arrows point in the same direction).
If the degree of the polynomial is odd and the leading coefficient is negative, then the end behavior is as follows:
As x approaches negative infinity, the graph of the polynomial goes up to positive infinity (the arrows point in the same direction).
As x approaches positive infinity, the graph of the polynomial goes down to negative infinity (the arrows point in the same direction).
For similar questions on polynomial
https://brainly.com/question/4142886
#SPJ11
MO
C
15 in
115
The triangle above has the following measures.
Use the 45 45 90 Triangle Theorem to find the
length of the hypolenuse Include correct units
Show all your work
The length of the hypotenuse of the triangle is b = x√2 units
Given data ,
Let the triangle be represented as ΔABC
And , the triangle is a 45 - 45 - 90 triangle
So , the two legs are congruent to one another and the non-right angles are both equal to 45 degrees
And , the sides are in the proportion x : x : x√2
Now , the length of the sides of the triangle are
The measure of base BC = a units = x
The measure of height AB = c units = x
So , the measure of the hypotenuse of the triangle is = x√2 units
Hence , the triangle is solved and hypotenuse AC = x√2 units
To learn more about triangles click :
https://brainly.com/question/16739377
#SPJ1
what is the end behavior of the function g(x)= -3e^x?
The end behavior of [tex]g(x) = -3e^x[/tex] is As x approaches positive infinity, g(x) approaches negative infinity, As x approaches negative infinity, g(x) approaches 0.
To determine the end behavior of the function g(x) = -3e^x, we need to consider what happens to the output of the function as x gets very large (approaches positive infinity) or very small (approaches negative infinity).
As x approaches positive infinity, e^x also approaches positive infinity, since e is a positive number greater than 1. Therefore, -3e^x approaches negative infinity as x approaches positive infinity. We can write this as:
lim x → ∞ (-3[tex]e^x[/tex]) = -∞
This means that the function g(x) approaches negative infinity as x approaches positive infinity.
Similarly, as x approaches negative infinity, e^x approaches 0, since e to a negative power is equal to 1 divided by e raised to the absolute value of that power. Therefore, -3e^x approaches 0 as x approaches negative infinity. We can write this as:
lim x → -∞ (-3[tex]e^x[/tex]) = 0
This means that the function g(x) approaches 0 as x approaches negative infinity.
So, in summary, the end behavior of g(x) = -3[tex]e^x[/tex] is:
As x approaches positive infinity, g(x) approaches negative infinity.
As x approaches negative infinity, g(x) approaches 0.
for such more question on negative infinity
https://brainly.com/question/22008756
#SPJ11
Find the missing side length.
Translate each graph as specified below.
(a) The graph of is shown. Translate it to get the graph of .
(b) The graph of is shown. Translate it to get the graph of .
look at picture
The graph of the translations are attached
The red is the preimage while the blue is the image
How to complete the translation
The absolute value graph has equation f(x) = |2x| and the translation we want to obtain is y = f(x) + 4 and this is a translation 4 units up
This results to a graph of equation y = f(x) + 4 = |2x| + 4
For the parabolic graph, the equation is written as
y = a(x - 2)² - 3
solving for a using (0, -5)
-5 = a(0 - 2)² - 3
-2 = 4a
a = -1/2, then the equation is g(x) = -1/2 (x - 2)² - 3
Applying the translation, the equation will be
y = g(x + 2) = -1/2 (x - 2 + 2)² - 3 = -1/2 (x)² - 3
Learn more about translation at
https://brainly.com/question/1046778
#SPJ1
In 2010, the population of a city was about 187,000. During the next 10 years, the population increased by about 1% each year. Write an exponential model that represents the population y of the city t years after 2010. Then estimate the population in 2020. Round your answer to the nearest thousand.
The exponential model for the population of the city t years after 2010 is given by:
y = 187,000 × 1.01^tThe population in 2020 can be estimated to be 187,000 × 1.01^10 = 200,270, which is rounded to 200,000.
Hope this helps! Have a nice day. :)[tex]\qquad \textit{Amount for Exponential Growth} \\\\ A=P(1 + r)^t\qquad \begin{cases} A=\textit{accumulated amount}\\ P=\textit{initial amount}\dotfill &187000\\ r=rate\to 1\%\to \frac{1}{100}\dotfill &0.01\\ t=years \end{cases} \\\\\\ A = 187000(1 + 0.01)^{t} \implies A=y = 187000(1.01)^t \\\\[-0.35em] ~\dotfill\\\\ \stackrel{\textit{in 2020 is 10 years later}}{y=187000(1.01)^{10}}\implies y\approx 207000[/tex]
Two square-shaped fields are next to each other. The perimeter of each field is 32 feet. The two fields are joined together to form a single rectangular field. What is the perimeter of the rectangular field? (1 point) a 40 feet b 48 feet c 56 feet d 64 feet
The solution is:
The area of the rectangular field is 2831.25 sq. yards.
The perimeter of the rectangular field is 153.1 yards
Here,
the length of the field = 62. 5 yards
The width of the field = 45.3 yards
PERIMETER OF A RECTANGLE = 2 (LENGTH + WIDTH)
⇒ The perimeter of the yard = 2 ( 62.5 + 45.3)
= 2 x (107.8) = 153.1 yards
or the perimeter of the rectangular field is 153.1 yards
Area of a Rectangle = (LENGTH x WIDTH)
⇒ The area of the yard = ( 62.5 x 45.3)
= 2831.25 sq. yards
or the area of the rectangular field is 2831.25 sq. yards.
To learn more on Area click:
brainly.com/question/20693059
#SPJ1
complete question:
John has a rectangular-shaped field whose length is 62.5 yards and width is 45.3 yards.
The area of the field is square yards. The perimeter of the field is yards.
Convert 11/15 15/11 to a decimal and a percent.
We can see here that converting 11/15 to decimal and a percent, we have:
Decimal = 0.7333 Percent = 73.33%Converting 15/11 to decimal and percent, we have:
Decimal = 1.3636Percent = 136.36%What is decimal?A decimal is a number that uses the decimal system, which is a system of representing numbers using the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9, and a decimal point.
Thus, converting 11/15 to decimal and a percent:
Decimal = 11 ÷ 15 = 0.7333 (rounded to four decimal places)Percent = 0.7333 × 100 = 73.33%Converting 15/11 to decimal and percent, we have:
Decimal = 15 ÷ 11 = 1.3636 (rounded to four decimal places)Percent = 1.3636 × 100 = 136.36%.Learn more about decimal on https://brainly.com/question/28393353
#SPJ1
I need help with this please
The features of the quadratic function are given as follows:
Zero: x = -2 with a multiplicity of 2.Axis of symmetry: x = -2.Minimum value of y = 0.Vertex of (-2,0).How to obtain the features of the quadratic function?The zero is the value of x for which the function touches or crosses the x-axis, hence it is of:
x = -2.
As the graph only touches the x-axis, the zero has an even multiplicity of 2.
The vertex is the turning point of the graph of the quadratic function, hence the coordinates are given as follows:
(-2,0).
Meaning that:
The axis of symmetry is of x = -2, which is the x-coordinate of the vertex.The function has a minimum value, as it is a concave up parabola, at y = 0.More can be learned about quadratic functions at https://brainly.com/question/31895757
#SPJ1
Name the polygon below. This is a(n) ____________.
a six-sided figure
septagon
pentagon
hexagon
octagon
Answer:
Hexagon
Step-by-step explanation: Count the sides, hex means 6, sept means 7, oct means 8
In ⊙C, DE≅FG. Select all the statements that must be true.
A. △DCE≅△FCE
B. arcDE arcFG
C. △FCE≅△FCG
D. ∠DCE≅∠GCF
E. EF≅ED
In a circle with congruent line segments DE and FG, statements A is true, B is false, C and D are false, and E may or may not be true.
In the given circle ⊙C, DE and FG are two line segments that are congruent. The following statements must be true:
A. △DCE≅△FCE: This statement is true because both triangles share the same side lengths of DE and FG and the common angle ∠DCE = ∠FCE.
B. arcDE = arcFG: This statement is false since congruent line segments do not necessarily correspond to congruent arcs in a circle.
C. △FCE≅△FCG: This statement is false since there is no given information to suggest that the line segment CG is congruent to CE.
D. ∠DCE≅∠GCF: This statement is false since there is no given information to suggest that ∠GCF is congruent to ∠DCE.
E. EF≅ED: This statement may or may not be true since we are not given any information about the length of either of these line segments.
Learn more on circle geometry here;
brainly.com/question/24375372
#SPJ1
Mr. Beachy's grade distribution over the past 3 years in Algebra 1 class is shown in the table below. Answer the following questions based on this table in simplest fraction form.
Grade # Students
A
41
B
183
C
265
D
96
F
85
Incomplete
2
How many total students are there?
If Sharon plans on taking Algebra 1 with Mr. Beachy, what is the empirical probability that she will receive an A?
If Cody plans on taking Algebra 1 with Mr. Beachy, what is the empirical probability that he will receive a B?
a.) The number of students that took the Algebra 1 for the whole three years = 672
b.) The empirical probability that Sharon will receive an A would be =0.061
How to calculate the number of students that took the Algebra 1?For question a.)
To calculate the total number of students that took the Algebra 1 for the 3 years, the students that got the various grades are being added up. That is;
Grade A= 41
Grade B =183
Grade C =265
Grade D =96
Grade F =85
Incomplete = 2
Total = 41+183+265+96+85+2 = 672
For b.)
The probability = possible outcome/sample space
possible outcome = 41/672=0.061
Learn more about probability here:
https://brainly.com/question/31123570
#SPJ1
A ship leaving dock A has to travel directly to a buoy (represented by point C below) 8.8 miles away. It then may proceed directly to dock B 10.5 miles away. The distance between dock A and dock B is 9.2 miles. What is the measure in degrees of the turn that the ship has to make at the buoy? Round your answer to the nearest tenth of a degree.
Answer:
17.7 degree
Step-by-step explanation:
We can begin by drawing a diagram:
```
C
/|
/ |
/α | 10.5 mi```
A------β------B
\ /
\ /
\/
8.8 mi
```
We can label angles β and α as shown, where β represents the angle from point C to point B and α```
A ------- 9.2 mi -------- B
```
Let's label some angles in the diagram:
- Angle A: the angle between the line from A to C and the line from A to B
- Angle B: the angle between the line from C to A and the line from C to B
- Angle α: the angle at the buoy C
We want to find the measure of angle α. To do that, we can use the law of cosines to find the length of the side opposite angle α:
```
c^2 = a^2 + b^2 - 2ab cos(C)
```
where a = 8.8, b = 10.5, and C = angle B. We can solve for cos(C):
```
cos(C) = (a^2 + b^2 - c^2) / 2ab
```
Since we know a, b, and c (8.8, 10.5, and 9.2 respectively), we can calculate cos(C):
```
cos(C) = (8.8^2 + 10.5^2 - 9.2^2) / (2 * 8.8 * 10.5) ≈ 0.701
```
Now we can find angle B using the inverse cosine function:
```
B = cos^-1(cos(C)) ≈ 45.4°
```
Finally, we can find angle α by subtracting angle A from angle B:
```
α = B - A ≈ 17.7°
```
So the measure of the turn that the ship has to make at the buoy is approximately 17.7 degrees.
4 miles in 25 minutes find the unit rate in miles/hour
Answer:
9.6 miles per hour
Step-by-step explanation:
The rate is 4/25 miles per minute
To convert to miles per hour, since there are 60 minutes in an hour, multiply: 4/25 x 60
= 240/25
= 9.6 miles per hour
What is the area of a right triangle with a height of 8 yards and a base of 17 yards?
A140¹ yds²
B136 yds²
C70 yds²
D35 yds²
The area of a right triangle is 70 yds².
We have,
Base= 17 yards
Height = 8 yards
The area of a triangle is given by the formula:
Area = (1/2) x base x height
Substituting the given values, we get:
Area = (1/2) x 17 x 8
Area = 68 square yards
Therefore, the area of a right triangle is 70 yds².
Learn more about Area of Triangle here:
https://brainly.com/question/19305981
#SPJ1
Here is the net of a squared-based pyramid. Calculate the height of a square-based pyramid.
The height if the squared-based pyramid is given as 34 cm
How to solve for the height of the shapeWe have BD / 2
= 10 / 2
= 5 cm
AC = [tex]\sqrt{AB^2-BC^2}[/tex]
= √12² - 5 ²
= 12 cm
The height is given as 10 + 12 + 12
= 34 cm
Hence the height if the squared-based pyramid is given as 34 cm
Read more on height here:https://brainly.com/question/1739912
#SPJ1
Which graph represents the compound inequality?
-3
O
O
O
-5-4-3-2-1 0 1 2 3 4 5
--5-4-3-2-1 0 1 2 3 4 5
--5-4-3-2-1 0 1 2 3 4 5
-5-4-3-2-1 0 1 2 3 4 5
A graph that represents the compound inequality include the following: D. graph D.
What is a number line?In Mathematics and Geometry, a number line simply refers to a type of graph with a graduated straight line which comprises both positive and negative numbers that are placed at equal intervals along its length.
This ultimately implies that, a number line primarily increases in numerical value towards the right from zero (0) and decreases in numerical value towards the left from zero (0).
Since the inequality symbol represents less than, we can logically deduce that the first point would be located at negative 3, followed by positive 1, with an open circle on both sides.
Read more on number line here: brainly.com/question/22515080
#SPJ1
Find the line that contains the
error, explain the error, and then
solve the problem correctly.
4+3(2x-3) = 6x - 5
Line 14+ 6x-9 = 6x-5
Line 2
10x-9 = 6x-5
Line 3-
4x = 4
Line 4-
x = 1
Answer:
Line 2
Step-by-step explanation:
In the second line, 4 is taken as 4x and added with 6x resulting in 10x.
The line 2 should be written as follows:
6x - 5 = 6x - 5
Transfer like terms to the same side of the equation.6x - 6x = 5 - 5
Subtract expressions.0 = 0
What is the end behavior of the graph of f(x) = -0.25x² - 2x +1?
A. As x increases, f(x) increases. As x decreases, f(x) decreases.
B. As x increases, f(x) decreases. As x decreases, f(x) decreases.
C. As x increases, f(x) increases. As x decreases, f(x) increases.
D. As x increases, f(x) decreases. As x decreases, f(x) increases.
What is the solution of 3√x+8 = -4? X =√72 O x=-66 O x = 58 O no solution
Answer: add image
Step-by-step explanation:
OORS 15200 In the beginning of BS. 2077, Dolma de posited B? 120,000 in her account at the rate of 9/. p.a. If she paids / of how her fnterest as income tax much total amount did she receive in mae beginning of BS 2080?
Okay, let's break this down step-by-step:
In the beginning of BS 2077, Dolma deposited B? 120,000 in her account.
The interest rate is 9% per annum.
So in the first year (BS 2077), she earned 9% of 120,000 = B? 10,800 in interest.
Her principal amount increased to 120,000 + 10,800 = B? 130,800
In BS 2078, she earned 9% of 130,800 = B? 11,772 in interest.
Her principal now was 130,800 + 11,772 = B? 142,572
In BS 2079, she earned 9% of 142,572 = B? 12,781 in interest.
Her principal now was 142,572 + 12,781 = B? 155,353
In the beginning of BS 2080, her total balance was B? 155,353.
She paid 6% of her interest as income tax.
9% of 155,353 is 13,991.
6% of 13,991 is 839.
So her total interest received in BS 2080 after paying tax was 13,991 - 839 = B? 13,152.
Her total amount received in BS 2080 was:
Principal (155,353) + Interest (13,152) = B? 168,505
So the total amount Dolma received in the beginning of BS 2080 was B? 168,505
Let me know if you have any other questions!
Which equation matches the graph of the greatest integer function below?
A. y = [x] - 4
B. y = [x] + 2
C. y = [x] - 2
D. y = [x] + 4
The equation that matched the graph of the greatest interger function is y = [x] - 4. The correct option is (A).
How to know the graph of an equationTo know the graph, we recall the equation of a straight line which is:
y = mx + c
where m is the gradient and
c is the intercept on the y-axis
Checking the graph, we can find the the intercept made by the straight line curve on the y-axis is -4.
Then we can have an equation of line written as:
y = mx + (-4) = mx - 4
From the options provided, there is only 1 with intercept of -4
Learn more about straight line here:
https://brainly.com/question/25969846
#SPJ1
Pls help!! How many more cups of lemonade would Jayla expect
to sell if the temperature was 150° F?
Using the model, at 150°F they can sell 304 cups.
How many can she expect to sell?We can see that the line that models this passes through (65, 35) and (80, 75)
Then the slope of the line is:
a = (75 - 35)/(80 - 65)
a = 2.66
y = 2.66*x + b
To find the value of b, we can use the fact that y = 35 when x = 65
35 = 2*65 + b
35 - 2*65 = b
-95
y = 2.66*x -95
IF the tempererature is 150°F, then:
y = 2.66*150 - 95
y = 304
They can expect to sell 304 cups.
And no, the 150° don't make sense, that is equivalent to 66°C.
LEarn more about lines:
https://brainly.com/question/25969846
#SPJ1
Solve part A on the picture
The mean of M1 + M2 is 1.4
The standard deviation is 1.104536.
How do we calculate?The mean of the number of mistakes = 0.7
The standard deviation of the number of mistakes = 0.781
(a) Mean of M1 + M2:
Mean of (M1 + M2)
= Mean(M1) + Mean(M2)
= μ + μ
= 2μ
= 2 * 0.7
mean of M1 + M2= 1.4
(b) Standard deviation of M1 + M2:
Standard deviation of (M1 + M2)
= √(Variance(M1) + Variance(M2))
Variance for (M1) = (σ²) = (0.781)² = 0.610561
Variance for (M2) = (σ²) = (0.781)² = 0.610561
Standard deviation for (M1 + M2)
= √(0.610561 + 0.610561)
= √1.221122
Standard deviation = 1.104536
Learn more about Standard deviation at:
https://brainly.com/question/24298037
#SPJ1
use substitution to solve the system y=3×-14 and 4x+5y=25
Answer:
x=58.74
Step-by-step explanation:
4x+5(-42)=25
4x+-210=25
4x=25+210
4x=235
x=235/4
x=58.75
pls help for brainliest!!! trig
Question 20:
the complex number in trigonometric form r(cos + i sin 8), with 8 in the interval [0". 360") is 3[cos 90+ i sin 90°).
Option B is correct.
Question 21.
The expression in the standard form a + bi.
[2(cos 15" + i sin 15°)] is [tex]3\sqrt{} 2 + 3\sqrt{2} i[/tex]
Option C is correct.
Question 22
The complex roots of-21 is :
5√2 (cos 54° + i sin 54°), %√2 (cos 126° + i sin 126°),% √2 (cos 198° + i sin 198°), 5√2 (cos 270° + i sin 270°), 5√2 (cos 342° + i sin 342°).
Option B is correct.
What is a complex root?Complex roots are described as the imaginary roots of equations, which are represented as complex numbers.
The expression : 2(cos 15° + i sin 15°) = 2e^(i15°) can be written in standard from as
2e^(i15°) = 2(cos 15° + i sin 15°)
= 2cos 15° + 2i sin 15°
=2cos 15° + 2i sin 15°
= [tex]3\sqrt{} 2 + 3\sqrt{2} i[/tex]
learn more about complex roots at:
https://brainly.com/question/29654300
#SPJ
Find the height of a tree that casts a 30-ft shadow when a 6-ft 6-in. pole casts a shadow of 3 ft.
The height of the tree is 60 ft
What is proportion?Proportion is defined as as an equation that is made equal to another
From the information given, we have that;
The shadow casted by the tree = 30 ft
The height of the pole = 6ft
The shadow casted by the pole = 3ft
Then, we have that;
If 6ft = 3ft
xft = 30ft
cross multiply the values, we have;
3x = 30(6)
Multiply the values
3x = 180
Divide both sides by the coefficient of the variable
x = 60ft
Learn more about proportion at: https://brainly.com/question/1781657
#SPJ1
3. There are shows scheduled for Theater A next week. All of the tickets for the seats have been sold. What is the expected attendance for next week?
Answer:
Initially, national security was defined as the government's ability to protect its citizens from military attacks. Today, this definition also includes other non-military areas such as defense against crime and terrorism, economic security, environmental security, food security, energy security and cyber security.
Use Fermats Little theorem to find 23^1002 mod 41
Answer:
231002 = (2340)25 · 232 ≡ 125 · 529 = 529 ≡ 37 mod 41.
1. Breifly discuss the applications of Matrix on word Problem Solution and on Marcov Chain rule, by taking specific examples.
Matrices are widely used in various fields, including solving word problems in mathematics because they provide a convenient way to organize and manipulate data, making complex problem-solving more manageable.
How can Matrix on word problem solutions be applied ?In word problem solutions, matrices serve as valuable tools for organizing and manipulating data, simplifying complex problem-solving endeavors. For instance, when faced with a system of linear equations, matrices provide a structured representation that enables efficient computation.
Consider a word problem involving multiple variables and equations, such as determining quantities and prices of diverse items. By formulating the problem as a matrix equation and employing matrix operations like Gaussian elimination or matrix inverses, we can decipher the unknown variables and unlock solutions.
Find out more on Matrix on word problems at https://brainly.com/question/29287013
#SPJ9
Find the quotient and remainder using synthetic division.
x^4
-
3
x^3
+
9
x
+
6/
x
+
1
The quotient is
The remainder is
The quotient of x⁴ - 3x³ + 9x + 6 ÷ x + 1 using synthetic division is x³ + 2x² - 2x + 7 and the remainder is 13
Finding the quotient and remainder using synthetic divisionFrom the question, we have the following parameters that can be used in our computation:
x⁴ - 3x³ + 9x + 6 ÷ x + 1
The synthetic set up is
-1 | 1 3 0 9 6
|__________
Bring down the first coefficient, which is 1 and repeat the process
-1 | 1 3 0 9 6
|__-1_-2_-2_7____
1 2 -2 7 13
This means that the quotient is
x³ + 2x² - 2x + 7
And the remainder is 13
Read more about synthetic division at
https://brainly.com/question/31664714
#SPJ1
Complete question
Find the quotient and remainder using synthetic division.
x^4-3x^3+9x+6/x+1
The quotient is
The remainder is
Answer:
Step-by-step explanation:
To use synthetic division, we need to set up the coefficients of the dividend polynomial in decreasing order of powers of x, including any missing terms with zero coefficients.
So we have:
Dividend: x^4 - 3x^3 + 9x + 6
Divisor: x + 1
We can represent the divisor as (x + 1) and set up the synthetic division table as follows:
-1 | 1 -3 9 0 6
| -1 4 -13 13
|___________________
| 1 -4 13 -13 19
The numbers on the first row of the table are the coefficients of the dividend polynomial in decreasing order of powers of x. The -1 on the left of the table is the opposite of the divisor, x + 1.
The first number in the second row is always 0, and we get it by bringing down the first coefficient, which is 1. To get the next number in the second row, we multiply the divisor, -1, by 1, and add the result to the second coefficient, which is -3. This gives us -1 x -3 = 3, which we write under -3 in the first row.
We repeat this process for the next numbers in the second row, always multiplying the divisor by the last number in the second row, and adding the result to the next coefficient in the first row. We continue until we reach the last coefficient in the first row.
The last number in the second row, 13, is the remainder of the division. The other numbers in the second row are the coefficients of the quotient polynomial in decreasing order of powers of x. So the quotient is:
x^3 - 4x^2 + 13x - 13
And the remainder is:
13
Therefore, the quotient is x^3 - 4x^2 + 13x - 13, and the remainder is 13.