Practice Questions

Question 1

if n^2is divisible by 72 and n is a positive integer. How do you find the largest positive integer that must divide n?

Answer 1

n^2 = 72k and 72k = perfect square.

n^2 = (2^3)(3^2)k
- k = 2m^2 [for 72k to be a perfect square]

n^2 = (2^4)(3^2)m^2

n = (2^2)(3)(m)

Positive integers that MUST divide n = 1, 2, 3, 4, 6, and 12.

12 is the largest positive integer that must divide n!

Question 2

163. Running at their respective rates, Machine X takes 2 days longer to product W widgets than Machine Y. At these rates, if the two machines together produce 5/4 w in 3 days, how many days would it take Machine X alone to produce 2w widgets?

Answer 2

12

Question 3

What is the greatest positive integer n such that 5^n divides 10! - (2)(5!^2)?

Answer 3

5

Question 4

The average length per film of a group of 21 films is t minutes. If a film that was 66 minutes long is replaced with a film that is 52 minutes long, in terms of t what is the average length per film now?

Answer 4

t - 2/3

Question 5

Finding the price of a shirt that is 12.5% decrease from 125

Answer 5

(112.5/100) x C = 125

Question 6

There are two positive integers an and b, where a > b. The sum of the squares of these integers is equal to 5 times their sum. The different of the squares of these integers is equal to 9 times their difference. What is the value of a - b?

Answer 6

This is a simultaneous equation! Set up the equations and subtract from one another.

Answer = 3

Question 7

For each positive integer k, let ak = 7k.

Which of the following is the greatest value of n such that 10^n divides (a1)(a2)(a3)….(a28)?

Answer 7

= 7^28 x 28

10 = 5 x 2

28/5 = 5
5/5 = 1

1 + 5 = 6 = n