Integer properties

Take X for calculation
x for which 57! / (7^x) is an integer
i.e X = 9 for which 57! / (7^x) is an integer

7x8 = 56 (Within 57, 56 highest number divisible by 7.)

Similarly
y for which 57! /(13^y) is an integer
i.e y = 4 for which 57! / (7^x) is an integer

13x4 = 52 (Withing 57, 52 highest number divisible by 13)


So if you check %change of X on Y = ((9-4)/(4))x100 = 125%

[email protected] wrote:If x and y are positive integers, the greatest value of x for which 57! / (7^x) is an integer is what percent greater than the greatest value of y for which
57! /(13^y) is an integer?

A. 71%

B. 100%

C.125%

D.225%

E. 250%

57! = 57*56*55*....*3*2*1.

To determine the greatest value of x for which 57!/(7^x) is an integer and the greatest value of y for which 57!/(13^y) is an integer, we need to count how many 7's and how many 13's can divide into 57!.
Put another way:
We need to count how many 7's and how many 13's are contained within 57!.
To count, simply divide increasing POWERS OF 7 AND 13 into 57.

Every multiple of 7 within 57! provides AT LEAST ONE 7:
57/7 = 8 --> eight 7's.
Every multiple of 7² within 57! provides a SECOND 7:
57/7² = 1 --> one more 7.
Thus, the total number of 7's contained within 57! = 8+1 = 9.
Result:
Since there are nine 7's contained within 57!, the greatest possible value of x for which 57!/(7^x) is an integer = 9.

Every multiple of 13 within 57! provides ONE 13:
57/13 = 4 --> four 13's.
Thus, the total number of 13's contained within 57! = 4.
Result:
Since there are four 13's contained within 57!, the greatest possible value of y for which 57!/(13^y) is an integer = 4.

Since 8 is 100% greater than 4, x=9 is A LITTLE MORE than 100% greater than y=4.

The correct answer is C.

For a similar problem, check here:
https://www.beatthegmat.com/how-to-find- ... 71343.html