What is the largest power of 6! that can divide 60! ?

Question

What is the largest power of 6! that can divide 60!?

A) 56
B) 42
C) 28 
D) 14
E) 7

souvonik2k · Accepted Answer

6!=6.5.4.3.2.1
6=3x2
The greatest prime in 6! is 5, so finding the maximum power of 5 in 60!
 \frac{60}{5}+\frac{60}{{5^2}}  = 12+2 = 14  
Answer D.

AKY13 · Answer

Hi, 
6fac has 2 to the power 4.

We need to find out how many 2 to the power 4 are present in 60fac.

60/2 = 30 
60/4 = 15
60/8 = 7 
60/16 = 3 
60/32 = NIL

Total 55 2's. 
55/4(2 is to the power of 4) = 13 (ignoring remainders)

Hence the answer must be 13 not 14.

Bismarck · Answer

AKY13

60/32 should be 1

AKY13 · Answer

Thanks for pointing out. However the answer is same, I think finding out the no. of 5s in 60 fac is not the correct way of doing. 2^4 is larger than 5 hence would be less in nos.

souvonik2k · Answer

Hi Similar question for practice: https://gmatclub.com/forum/what-is-the- ... fl=similar For theory you can refer to the following blog: https://www.gmatclub.com/forum/veritas-prep-resource-links-no-longer-available-399979.html#/2011/0 ... actorials/ Hope it helps.

ScottTargetTestPrep · Answer

We note that 6! = 6 x 5 x 4 x 3 x 2 x 1 = 5^1 x 3^2 x 2^4 = 16 x 9 x 5. We need to determine the greatest power of these factors present in 60!.

To determine the number of 16s within 60!, we need to determine the number of 2s. We can use the following shortcut in which we divide 60 by 2, then divide the quotient of 60/2 by 2 and continue this process until we no longer get a nonzero quotient.

60/2 = 30

30/2 = 15

15/2 = 7 (we can ignore the remainder)

7/2 = 3 (we can ignore the remainder)

3/2 = 1 (we can ignore the remainder)

Since 1/2 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 2 within 60!.

Thus, there are 30 + 15 + 7 + 3 + 1 = 56 factors of 2 within 60!. Since 16 = 2^4, the largest power of 16 that divides 60! is 14.

Doing the same for the prime factor of 3 to determine the number of 9s in 60!, we find that the largest power of 9 that divides 60! is also 14.

Finally, since there are 12 multiples of 5 in 60! and both 25 and 50 contains two 5’s apiece, we see that if we were to break 60! into primes, we’d have 14 factors of 5. Thus, the largest power of 6! that divides 60! is 14.

Answer: D

NitishJain · Answer

I solved this using number of 0's in 60!
6! can be written as 2^4*3^2*5 : It will has only 0 which depends on power of 5
So i counted that 60! will have 5^14

so 14

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What is the largest power of 6! that can divide 60! ?

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