Number Theory : Division of factorials, remainders

The highest power of 10 that can divide a factorial. Number of trailing zeroes.

Question

A person starts multiplying consecutive positive integers from 20. How many numbers should he multiply before the will have result that will end with 3 zeroes?

1. 11
2. 10
3. 6
4. 5
   

Correct Choice is (3) and correct answer is 6

Explanatory Answer

A number will end in 3 zeroes when it is multiplied by 3 10s.
To get a 10, one needs a 5 and a 2.

Therefore, this person should multiply till he encounters three 5s and three 2s.
20 has one 5 (5 * 4) and 25 has two 5s (5 * 5).
20 has two 2s (5 * 2 * 2) and 22 has one 2 (11 * 2).

Therefore, he has to multiply till 25 to get three 5s and three 2s, that will make three 10s.
So, he has to multiply from 20 to 25 i.e. 6 numbers.

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