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m+2=10^n

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by outreach » Fri Jul 02, 2010 10:12 am
If m+2=10^n, where m and n are positive integers, n=?
(1) The sum of digits of m is 116.
(2) n>10
A. Statement (1) ALONE is sufficient but Statement (2) ALONE is not sufficient.
B. Statement (2) ALONE is sufficient but Statement (1) ALONE is not sufficient.
C. BOTH Statements TOGETHER are sufficient, but NEITHER Statement alone is sufficient.
D. Each Statement ALONE is sufficient.
E. Statements (1) and (2) TOGETHER are NOT sufficient.
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by albatross86 » Fri Jul 02, 2010 10:26 am
m + 2 = 10^n; m > 0, n > 0

What is n?

Statement 1:

The sum of the digits of m = 116

In order for n to be a positive integer, m+2 must be a power of 10.

Thus m has to be a power of 10, minus 2.

Egs. If n = 1, m = 8
n = 2, m = 98
n = 3, m = 998
n = 4, m = 9998

Thus m will always be of the form of a number of 9's followed by a single 8.

Subtract this single 8 from the sum of the digits => 116 - 8 = 108

Divide this by 9 to get the number of 9's => 108/9 = 12

So m = 9999999999998 (Irrelevant to right it down, but it's kinda fun to type long numbers :D)

Thus n = 13 (No need to calculate)

SUFFICIENT

Statement 2:

n > 10

INSUFFICIENT.

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