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# If N is the product of all integers from 1 to 100, both inclusive ....

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e-GMAT Representative
Joined: 04 Jan 2015
Posts: 2432
If N is the product of all integers from 1 to 100, both inclusive ....  [#permalink]

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Updated on: 20 Dec 2018, 03:26
00:00

Difficulty:

75% (hard)

Question Stats:

57% (01:25) correct 43% (01:57) wrong based on 74 sessions

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If N is the product of all integers from 1 to 100, both inclusive, then what is the remainder when N + 100 is divided by $$7^{16}$$?

A. 0
B. 7
C. 49
D. 100
E. 1343

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Originally posted by EgmatQuantExpert on 19 Dec 2018, 10:22.
Last edited by EgmatQuantExpert on 20 Dec 2018, 03:26, edited 1 time in total.
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Joined: 02 Aug 2009
Posts: 7199
Re: If N is the product of all integers from 1 to 100, both inclusive ....  [#permalink]

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19 Dec 2018, 23:20
EgmatQuantExpert wrote:
If N is the product of all integers from 1 to 100, both inclusive, then what is the remainder when N + 100 is divided by $$7^{16}$$?

A. 0
B. 7
C. 49
D. 100
E. 1343

N is divisible by all numbers till 100, so let us check how many 7s are there...
[100/7]+[100/7^2]=14+2=16, so N is divisible by 7^(16).
N is divisible, so let us check for 100..
7^(16) is >100, so 100 will be the remainder..

D
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Re: If N is the product of all integers from 1 to 100, both inclusive ....  [#permalink]

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01 Jan 2019, 21:52

Solution

Given:
We are given that,
• N = 1 * 2 * 3 * 4 * … * 99 * 100

To find:
We are asked to find out,
• The remainder when N + 100 is divided by $$7^{16}$$

Approach and Working:
• We know that,$$R{\frac{(N + 100)}{7^{16}}} = R(\frac{N}{7^{16}}) + R(\frac{100}{7^{16}})$$
o R$$(\frac{N}{7^{16}}) = R\frac{(1 * 2 * 3 * 4 * .... * 99 * 100)}{7^{16}} = R(\frac{100!}{7^{16}})$$

• Power of 7 in 100! = $$[\frac{100}{7}] +[\frac{100}{7^2}] = 14 + 2 = 16$$
o So, $$100! = 7^{16} * k$$, where k is an integer

• Thus, $$R(\frac{100!}{7^{16}}) = 0$$, and
• $$R(\frac{100}{7^{16}}) = 100$$

Therefore, the remainder when N + 100 is divided by $$7^{16}$$ is 0 + 100 = 100

Hence the correct answer is Option D.

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Re: If N is the product of all integers from 1 to 100, both inclusive .... &nbs [#permalink] 01 Jan 2019, 21:52
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