Find the least number n such that no factorial has n trailing zeroes,

Question

Find the least positive integer n such that no positive integer factorial has n trailing zeroes, or n + 1 trailing zeroes or n + 2 trailing zeroes.

A. 624
B. 153
C. 126
D. 100
E. 18

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JerryAtDreamScore · Answer

Breaking Down the Info:

As our factorials get higher, the number of trailing zeros would get higher and the number would start jumping up when we reach certain thresholds.

For example, from 99! to 100! we created 2 more trailing zeros, hence a 2-number jump. We would like to know the lowest number of trailing zeros needed for a 4-number jump.

To create a trailing zero, we need a factor of 2 and a factor of 5. The factor of 5 is much less common than the factor of 2, so we only need to gather factors of 5's to find the number of trailing zeros.

Therefore using the logic above, we will need  5^4  as the next factorial so that we instantly jump 4 trailing zeros.  5^4 = 625 . Hence 625! is our threshold.

Finally, we need to count the number of trailing zeros in 625 or the number of multiples of 5s. We can count the layers to make this easier:

Multiple of 5's: 625/5 = 125.

Multiple of 25's: 625/25 = 25

Multiple of 125's: 625/125 = 5.

Multiple of 625's: 625/625 = 1.

Adding everything up gives us 156. Our count of trailing zeros jumped from 152 to 156 with 624! to 625!, then n here is 153.

Answer: B

Kinshook · Answer

Find the least positive integer n such that no positive integer factorial has n trailing zeroes, or n + 1 trailing zeroes or n + 2 trailing zeroes.

Number of trailing zeros for 124! = 24 + 4 = 28
Number of trailing zeros for 125! = 25 + 5 + 1 = 31
There are no trailing zeros of 29 & 30 in numbers

Number of trailing zeros for 624! = 124 + 24 + 4 = 152
Number of trailing zeros for 625! = 125 + 25 + 5 + 1 = 156
There are no trailing zeros of 153, 154 & 155 in numbers
n = 153

IMO B

zlishz · Answer

JerryAtDreamScore  can you please explain the thought process behind identifying that we need to find the lowest number of trailing zeroes to get a  4 number jumper ?

rak08 · Answer

Bunuel 
 KarishmaB

can you explain what the question exactly means?

KarishmaB · Answer

It is based on the concept of Highest Power in Factorial. Look here: https://anaprep.com/number-properties-h ... actorials/

A trailing zero (0 at the end of a number) is obtained when the number has 10 as a factor. A 10 needs a 2 and a 5. There are typically enough 2s but fewer 5s in factorials so number of 5s in a factorial give us the number of trailing zeroes.

5! has 1 trailing zero because it has exactly one 5 in it.
6! has 1 trailing zero because it has exactly one 5 in it.
...
9! has 1 trailing zero because it has exactly one 5 in it.
10! has 2 trailing zeroes because it has 2 5s in it.
24! has 4 trailing zeroes because it has 4 5s in it.
25! has 6 trailing zeroes because it has 6 5s in it. (25 brings in 2 extra 5s)

There is no number which has 5 trailing zeroes.

The question asks for the smallest number n such that no positive integer factorial has n trailing zeroes, or n + 1 trailing zeroes or n + 2 trailing zeroes.
So a number that introduces 4 5s will be required. That smallest number is 625 (which brings in 4 5s)

So 624! has 152 5s in it. 625! will have 156 5s in it.
So no factorial has 153, 154 and 155 5s in it. No factorial has n, n+1 and n+2 5s)

rak08

Bunuel
KarishmaB

can you explain what the question exactly means?

Find the least number n such that no factorial has n trailing zeroes,

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Find the least number n such that no factorial has n trailing zeroes,

Prep Toolkit

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