If $a$ and $b$ are positive integers for which $ab - 6a + 5b = 373$, what is the minimal possible value of $|a - b|$?

Answers

Answer 1

Answer:

  31

Step-by-step explanation:

We can solve the given equation for 'b', then find the integer values of 'a' that make 'b' a positive integer. There are 3 such values. One of these minimizes the objective function.

__

solve for b

  ab +5b = 373 +6a . . . . . . isolate b terms by adding 6a

  b = (6a +373)/(a +5) . . . . . divide by the coefficient of b

  b = 6 +343/(a +5) . . . . . . . find quotient and remainder

integer solutions

The value of 'b' will only be an integer when (a+5) is a factor of 343. The divisors of 343 = 7³ are {1, 7, 49, 343}. so these are the possible values of a+5. Since a > 0, we must eliminate a+5=1. That leaves ...

  a = {7, 49, 343} -5 = {2, 44, 338}.

Possible values of b are ...

  b = 6 +343/{7, 49, 343} = 6 +{49, 7, 1} = {55, 13, 7}

Then possible (a, b) pairs are ...

  (a, b) = {(2, 55), (44, 13), (338, 7)}

objective function

The values of the objective function for these pairs are ...

  |a -b| = |2 -55| = 53

 |a -b| = |44 -13| = 31 . . . . . the minimum value of the objective function

  |a -b| = |338 -7| = 331


Related Questions

Given that the expression “p dollars for every q items” describes a unit price, which statement must be true?

Answers

Hello, wheres the statement? Please provide full questions

Four students are trying to determine what number must replace the question (x^2)^?=x^4*x^8 in order to make it true
Joe said the answer is 6. Sam said it is impossible to answer.
Peter said the answer is 32. Alex said the answer is 16.
Who is correct? In words explain why.

Answers

Joe is correct as the unknown number is 6.

What are exponents?

The exponents of a number are defined as the representation of a number that shows how many times a number is multiplied by itself.

Given :

[tex](x^2)^?=x^4\times x^8[/tex]

Let the question mark exponent outside the bracket be y, which needs to be determined

[tex](x^2)^y=x^4\times x^8[/tex]

Simplify the equation by applying exponent rules to either side of the equation:

Apply the rule [tex](a^b)^c = a^{b \times c}[/tex]

[tex](x^2)^y=x^4\times x^8[/tex]

[tex](x)^{2y}=x^4\times x^8[/tex]

Now apply [tex]a^b \times a^c = a^{b+c}[/tex]

[tex](x)^{2y}=x^{4+8}\\\\(x)^{2y}=x^{12}[/tex]

So, the base is the same the exponent will be the same also,

2y = 12

y = 6

Hence, Joe is correct as the unknown number is 6.

Learn more about exponentiation here:

https://brainly.com/question/26938318

#SPJ1

The frequency table shows the results asking people how many hours they spend online per week

Answers

The histogram is attached with the answer.

What is a Histogram ?

It is a mathematical representation of data in a bar shaped graph , the range is user defined.

The histogram is drawn and the shape of the histogram is left skewed as the peak of the graph lies to the left of the centre.

The histogram is attached with the answer.

The complete question is

the frequency table shows the results of a survey that asked people how many hours they spend working in the yard per month. display the data in a histogram. describe the shape of the distribution

The data is attached with the answer.

To know more about Histogram

https://brainly.com/question/16819077

#SPJ1

In a family the ages of children arranged from youngest to oldest are 2,6,y,14 and z if the square of ages is in an a.p what is the ratio of z and y

Answers

Answer:

y=10

z=18

Step-by-step explanation:

2,6,y,14,z

there are 4 years different between the childrens.

2-6-10-14-18

Which expression is equivalent to?

Answers

The expression obtained is 2g³-5g² +6 , Option C is the correct answer.

What is an Expression ?

An expression is a mathematical statement that consists of constant , variable and mathematical operators.

The expression given is

[tex]\rm \dfrac{(5g^4 +5g^3 -17g^2 +6g) -(3g^4 +6g^3-7g^2 -12)}{g+2}[/tex]

On subtracting the numerator

2g⁴-g³-10g²+6g+12

Now dividing the numerator by denominator

The expression obtained is 2g³-5g² +6

Therefore Option C is the correct answer.

To know more about Expression

https://brainly.com/question/14083225

#SPJ1

Quick
Check
Match each power of a power expression with its simplified expression.

Answers

(4^-3)^-3 =4^9 (4^6)^-3 =1/4^18. (4^0)^-9 =4^0. (-4^9)^2 =4^18

can someone please help mee (20 points and i will give brainliest!!!)

Answers

Answer:

Step-by-step explanation:

1kg of sugar costs ₹25.30 . Find the cost of 13.5kg of sugar

Answers

Answer:

₹ 341.55

Step-by-step explanation:

Cost of 1 kg sugar = ₹ 25.30

Cost of 13.5 kg sugar = 25.30 × 13.5 = 341.55

Hence, the cost of 13.5 kg is ₹ 341.55

If you are writing, write statements too as in my school they give extra marks.

Thankyou and have a great day ahead.

Answer:341.55

Step-by-step explanation:

If one KG of sugar does cost  ₹25.3 and if you want to know how much 13.5 kg costs you just would have to multiply/times  ₹25.3 by 13.5 and what is 25.3 x 13.5.  ₹341.55 and that's your answer!

Venn diagram ( attached image )

Answers

Assessing the given Venn diagram and the data related to it, we can answer the required question:

Q1. How many people were surveyed = 86.

Q2. What is the probability that a customer chosen at random:

a) takes salt P(S) = 46/85.

b) takes both salt and vinegar P(S ∩ V) = 13/85.

c) takes salt or vinegar or both P(S ∪ V) = 66/85.

The number of people who take salt chips n(S) = 46.

The number of people who take vinegar chips n(V) = 33.

The number of people who takes both n(S ∩ V) = 13.

The number of people who take neither n((S ∪ V)') = 19.

Therefore, the total number of people surveyed n(U) = n(S) + n(V) - n(S ∩ V) + n((S ∪ V)') = 46 + 33 - 13 + 19 = 85.

The number of people who takes salt or vinegar or both n(S ∪ V) = n(U) - n((S ∪ V)') = 85 - 19 = 66.

The probability that a customer chosen at random takes salt P(S) = n(S)/n(U) = 46/85.

The probability that a customer chosen at random takes both salt and vinegar P(S ∩ V) = n(S ∩ V)/n(U) = 13/85.

The probability that a customer chosen at random takes salt or vinegar or both P(S ∪ V) = n(S ∪ V)/n(U) = 66/85.

Learn more about Venn Diagrams at

https://brainly.com/question/24713052

#SPJ10

Estimate
12.4 × 18
—————-
0.012

Answers

Answer: 12.4 x 18

-------------- = 18600

0.012

Work out (6 × 10²) ÷ (3 × 105)
Give your answer in standard form.

Answers

Answer: 40/21

Step-by-step explanation:

[tex]3 \times 105=315\\\\6 \times 10^{2}=600\\\\\implies \frac{6 \times 10^{2}}{3 \times 105}=\frac{600}{305}=\boxed{\frac{40}{21}}[/tex]

evaluate the expression below when x is 4

3x^2 +5 =​

Answers

Answer:

[tex]\fbox {53}[/tex]

Step-by-step explanation:

⇒ 3x² + 5

Substitute x = 4 :

⇒ 3(4)² + 5

⇒ 3(16) + 5

⇒ 48 + 5

53

3x^2+5
3(4)^2+5
*note*
(4)^2= 4•4=16
3(16)+5
48+5= 53

P(x)⋅Q(x)=R(x); if P(x)=x+1 and R(x)=2x3+2x2+3x+3, what is Q(x)?
A. 2x2+2
B. 2x2+3
C. 2x2−3
D. 2x2−2

Answers

Answer: B

Step-by-step explanation:

[tex]P(x) Q(x)=R(x) \implies Q(x)=\frac{R(x)}{P(x)}\\\\Q(x)=\frac{2x^{3}+2x^{2}+3x+3}{x+1}={2x^{2}(x+1)+3(x+1)}{x+1}=\boxed{2x^{2}+3}[/tex]

(6 + 7) + 8 = 21

Choose an equivalent addition sentence that shows the associative property of addition.

Answers

Answer:

6 + (7 + 8) = 21

Step-by-step explanation:

the associative property of addition is: (a + b) + c = a + (b + c)

So the equivalent addition sentence would be:

(6 + 7) + 8 = 6 + (7 + 8) as they both equal 21

Solve for the value of x:-
4x+3(x-1)=67​

Answers

Answer:

x = 10

Step-by-step explanation:

4x + 3(x - 1) = 67 ← distribute parenthesis and simplify left side

4x + 3x - 3 = 67

7x - 3 = 67 ( add 3 to both sides )

7x = 70 ( divide both sides by 7 )

x = 10

Hi. . .

✦•···················•✦•···················•✦

A n s w e r : -

[tex]\boxed{\fbox{\huge{x = 10}}}[/tex]

E x p l a n a t i o n :

Let's solve your equation step-by-step ~

➳ Step 1: Simplify both sides of the equation.

[tex]\sf{4x \: + \: 3(x \: - \: 1) = 67}[/tex]

[tex]\sf{4x \: + \: (3)(x) \: + \: (3)(-1) = 67} \: (Distribute)[/tex]

[tex]\sf{4x \: + \: 3x \: + \: -3 = 67}[/tex]

[tex]\sf{(4x \: + \: 3x) \: + \: (-3) = 67} \: (Combine \: like \: terms)[/tex]

[tex]\sf{7x \: + \: 7 \: -3 = 67}[/tex]

[tex]\sf{7x \: - \: 3 = 67}[/tex]

➺  Step 2: Add 3 to both sides.

[tex]\sf{7x \: - \: 3 \: + \: 3 = 67 \: + 3}[/tex]

[tex]\sf{7x = 70}[/tex]

➽ Step 3: Divide both sides by 7.

[tex]\sf{\frac{7x}{7} = \frac{70}{7}[/tex]

[tex]\boxed{\fbox{\huge{x = 10}}}[/tex]

Hope It Helps...

In a school, the number of girls is 70 more than boys. the total number of students is 1280. find the number of girls and boys simple equations

Answers

Answer:

Boys = 605

Girls = 675

Step-by-step explanation:

Let x be the number of boys in school.

So, girls = (x+70)

Total no. of students = 1280

x + x + 70 = 1280

2x + 70 = 1280

2x = 1280 - 70

2x = 1210

x = 1210/2

x = 605

Hence, no. of boys = 605

no. of girls = 605 + 70 = 675

What is the quotient of 0.8449 ÷ 0.71?

Answers

Answer:

1.19

Step-by-step explanation:

1) Shift the decimals of both 0.8449 and 0.71 to the right by 2 decimal places. (This is needed to convert 0.71 to 71, since long division must divide by a whole number.)

84.49 ÷ 71

2)  Use the algorithm method.

1 . 1 9

7 1 | 8 4 . 4 9

7     1 .  

1 3 . 4

7 . 1

6 . 3 9

6 . 3 9

3) Therefore,  0.8449 ÷ 0.71 = 1.19.

1.19

HELPPPPPPPPPPPPPPPPP PLEASE

Answers

Y=-4
X=2
Yeah that’s all I got to say lol

775 tickets were sold for a game for a total of $1,587.50. If adult tickets sold for $2.50 and childrens tickets sold for $1.50,how many of each kind of tickets were sold?

Answers

350 tickets sold of children and 425 tickets sold for adult.

What is algebra?

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols.

Let x be the number of adult tickets that were sold and y be the number of children's tickets that were sold.

Total 775 tickets were sold for the game.

So,

x + y = 775

If adults tickets sold for 2.50 and children's tickets sold for 1.50 and all the tickets were sold for a total of 1587.5, it means that

2.5x + 1.5y = 1587.5

put x = 775 - y

2.5(775 - y) + 1.5y = 1587.5

1937.5 - 2.5y + 1.5y = 1587.5

-y = 1587.5- 1937.5

y = 350

and, x = 775 - y = 775 - 350

x = 425

Learn more about this concept here:

https://brainly.com/question/15371020

#SPJ1

Find the nature of the graph of the function y = -9x3 + 5x2 - 2x using end behavior.

Answers

The graph goes upwards on the left side and downwards on the right side, so the correct option is the first one.

How is the end behavior of the function?

We have the function:

[tex]y = -9*x^3 + 5*x^2 -2x[/tex]

This is a cubic polynomial with a negative leading coefficient.

So, when x takes large negative values, because the leading coefficient is negative, y will take large positive values.

And when x takes large positive values, because the leading coefficient is negative, y will take large negative values.

So the nature is:

The graph goes upwards on the left side and downwards on the right side. The correct option is the first one.

If you want to learn more about end behavior:

https://brainly.com/question/1365136

#SPJ1

What is sin and cos ​

Answers

    Sin and cos are trigonometry functions used to find measurements in a right triangle.

         [tex]\displaystyle sin=\frac{\text{opposite side}}{hypotenuse}[/tex]

         [tex]\displaystyle cos=\frac{\text{adjacent side}}{hypotenuse}[/tex]

find x and y. Please help me with my math

Answers

Answer: Read attachment.

Step-by-step explanation: Attached below is my work. Hope it helps!

30 point!!!! Help me fast!!!PLS!!Question is attached!!!

Answers

Answer:

This is an extremely easy question bro!

1.

2. 3 (Base) ² (Exponent)

3. Expanded form: (9 * 1)

   Standard form: 9

Step-by-step explanation:

Well, there's none.

Please mark me as Brainliest! Thanks! :)
Detailed Answers:

1) 5^3

2) Base = 5; Exponent = 2

3) Expanded Form = 5 * 5
Standard Form = 25

It takes 6 hours for 2 people to build a swing set. Can you use the proportion 2 - 6 = 5 - h to determine the number of hours h it will take 5 people to build the swing set? Explain.

Answers

The number of hours, h it will take 5 people to build the swing set is; 2.4 hours

What is the number of hours it will take 5 people to build the swing set?

From the task content, it follows that it took 6 hours for 2 people to build the swing set, hence, it follows that it'd take 12 hours for 1 person to build the swing.

However, when the work is divided between 5 People, the time taken is therefore; 12/5 = 2.4hours.

Read more on proportionality;

https://brainly.com/question/24868934

#SPJ1

In order to create a system of equations that has the same solution as the system
x+y=4
2x+y = 7
complete three steps.
1
Step 1: Multiply the equation x + y = 4 by _[blank A]__.
Step 2: Add the equations 2x + y = 7 and _[blank B]_.
Step 3: Write the new system of equations as _[blank C]__.
Match each blank with the system or statement that correctly fills in that blank.

Answers

Blank A is -2. Blank B is -2x-2y=-8. Blank C is y=1

#Step1

Multiply by -2

It will result -2x-2y=-8

#2

-2x-2y=-8

#3

add

-y=-1y=1

Find x

x+1=4x=3

A bakery has a sale on cookies. If a customer buys a dozen cookies for $15, each additional single cookie is $0.75. What is an equation that shows how the total cost depends on the number of single cookies purchased? (Assume that the order starts with a dozen cookies.) How much would a dozen cookies plus six single cookies cost?

Answers

Answer:

T = 15 + 0.75c

T = $19.5

Step-by-step explanation:

Assume that T stands for total cost and c stands for number of single cookies purchased, the equation would be:

T = 15 + 0.75c

Therefore, a dozen cookies plus six single cookies would cost:

T = 15 + 0.75(6)

T = 15 + 4.5

T = $19.5

How many solutions does the system have?
x + y = 5
x + y = 6
Onone
O one
O two
O infinite

Answers

Answer:

none

Step-by-step explanation:

please mark me as brainlest

None because the interfere

Find the length of diagonal BC of ABCD to the nearest hundredth.

Answers

The length of BC as shown in the diagram is 6.32units

Calculating distance between two points

The formula for calculating the distance between two points is expressed as:

D = √(x2-x1)²+(y2-y1)²

Given the following coordinate points (3, 6) and (5, 0)

Substitute

BC = √(0-6)²+(5-3)²

BC = √(36+4)

BC = √40

BC = 6.32 units

Hence the length of BC as shown in the diagram is 6.32units

Learn more on distance formula here: https://brainly.com/question/661229

#SPJ1

What is the area of triangle ABC?

Answers

Answer:

3 square units

Step-by-step explanation:

Use BC as the base   =   2

   Use y -axis to C   (or B) as the height =  3

Area of triangle = 1/2  * base * height = 1/2 * 2 * 3 = 3 units^2

* GEOMETRY

please help me show the work for 9, 10, and 11!!

Answers

9) Diagonals of a parallelogram bisect each other, so KN=NM. Thus, KM=13.5+13.5=27 cm

10) Opposite sides of a parallelogram are congruent, so LM=KJ=17 cm

11) Based on Question 9, MN=13.5 cm

Other Questions
Type the correct answer in each box. Use numerals instead of words. Function g is a transformation of function f. Graph shows 2 exponential functions. First curve f enters quadrant 3 at (minus 6, minus 2) rises through (0, minus 1) and (1, 0) and (2, 2) in quadrant 1. Second curve g enters quadrant 2 at (minus 6, 6) falls through (0, 3) and (1, 0) in quadrant 1. What is the equation of function g? g(x) = f(x) Which of the following is written in the first-person point of view?A) When I am dead, my dearest, sing no sad songs for me.OB) The singer closed her eyes and took a deep breath.OC) A hush fell over the audience.D) No one expected what happened next. Which it it? what should you consider when choosing a type of activity Knowledge of which of the followingtypes of funding is required for anagricultural management position?A.CrowdfundingB. Micro loansC. Vertical integrationD. Government grants at what height will the air pressure equal 50% NEED THIS ASAP! Britain, with an army to enforce her tyranny, has declared that she has a right (not only to TAX) but to BIND us in ALL CASES WHATSOEVER, and if being bound in that manner, is not slavery, then is there not such a thing as slavery upon earth. . . .. . . Neither have I so much of the infidel in me, as to suppose that He has relinquished the government of the world, and given us up to the care of devils; and as I do not, I cannot see on what grounds the king of Britain can look up to heaven for help against us: a common murderer, a highwayman, or a house-breaker, has as good a pretence as he.The words slavery, devils, and murderer evoke a sense ofA.anger and injustice.B. fear and anxiety.C. confusion and chaos.D. guilt and shame. Why was James Wright important to Georgia during the late 1770s?A. He fought for Georgia in the Battle of Kettle Creek.B. He helped Georgia with strategy for the siege of Savannah.C. He expressed the feelings of the people of Georgia to Britain.D. He was sent to Georgia to restore the colony to the British Crown. . If we start with one bacterium and it doubles innumber every 15 minutes, HOW LONG WILL ITTAKE for one bacterium to become one million?? Which expression is equivalent to (startfraction 4 m n over m superscript negative 2 baseline n superscript 6 baseline endfraction) superscript negative 2? assume m not-equals 0, n not-equals 0. Pls tell me how to solve this! 7 on the pH scale represents what? A satellite completes one revolution of a planet in almost exactly one hour. At the end of one hour, the satellite has traveled 2.0 107 meters and is only 10 meters away from its starting point. What is the numerical value of the satellite's average velocity after that one hour? Can somebody help me, i would really appreciate it!! The price of a pack of kitchen rolls is reduced by 1/6The new price is 1.20Work out the original price. Hi people! Can you help me figure this out? I'll mark brainliest! If n= 1999 ! then [tex]\large \sf\sum\limits_{x = 1}^{1999} log_{n}(x) [/tex] is equal to a) -1b) 0c) 1d) [tex] \sf \sqrt[1999]{1999} [/tex] In what ways are the Navajo stories in conflict or consonance with historical explanations of migrations from far north of the continent to Dinetah in the Southwest? Jeff and Ryan rent a house from Mike and Yolanda Reynolds. They are three months behind in their rent and neighbors have been complaining about the noise coming from the home. Mike and Yolanda have tried to work with Jeff and Ryan to help them get back on track with the tardy rent, but the guys are not cooperating. Mike and Yolanda have decided they have no choice but to start the eviction process. Mike wants to just go in and remove the possessions, shut off all the utilities, and change the locks. Yolanda believes Mike's idea may be illegal and thinks there are other options. Is she correct and, if so, what are the options A 12 foot ladder leans against a wall so that the base of the ladder is 7 feet from the wall. How high up on the wall will the ladder reach? Round to the nearest hundredth of a foot. When division of labor is extensive, a. employees are often highly challenged. b. employees specialize in many tasks. c. jobs tend to be large. d. employees specialize in a single task. e. employees are often inefficient.