What is the largest value of non-negative integer N for

Hi All,

We're asked for the largest value of N for which 10^N is a factor of 50! This question comes down to 'prime factorization' and requires a degree of thoroughness on your part (to make sure that you find all of the 10s in 50!)

To start, 50! = (50)(49)(48)(47)(46)(45).....(3)(2)(1), so 50! is a really big number. You do not have to calculate it though; you just have to find all of the 10s that are hidden in that product.

10 = (2)(5), so we should start by looking for all of the multiples of 5 in 50!...
5, 10, 15 and 20 each contain a "5"
25 actually contains TWO 5s (re: 5x5)
30, 35, 40 and 45 each contain a "5"
50 contains TWO 5s (re: 2x5x5)

Thus, there are 4 + 2 + 4 + 2 = TWELVE 5s

Each of those 5s can be multiplied by a 2 (and there are LOTS of 2s in 50!), so we'll end up with twelve 10s. Thus, the maximum possible value of N is 12.

Final Answer: C

GMAT assassins aren't born, they're made,
Rich

swerve wrote:What is the largest value of non-negative integer N for which 10^N is a factor of 50!?

A. 5
B. 6
C. 12
D. 15
E. 20

The OA is C

Source: GMAT Prep

To determine the largest value of N, we need to determine how many times 10 divides 50!. Since 10 breaks into primes of 5 and 2, and since there are there are fewer 5s in 50! than 2s, we can find the number of 5s and thus be able to determine the number of 5-and-2 pairs.

To determine the number of 5s within 50!, we can use the following shortcut in which we divide 50 by 5, then divide the quotient of 50/5 by 5 and continue this process until we no longer get a nonzero quotient.

50/5 = 10

10/5 = 2

Since 2/5 does not produce a nonzero quotient, we can stop.

The final step is to add up our quotients; that sum represents the number of factors of 5 within 50!.

Thus, there are 10 + 2 = 12 factors of 5 in 50! This also indicates that there are twelve 5-and-2 pairs, and hence there are 12 factors of 10 within 50!

Answer: C