If n is a Non-Negative Integer, is \(10^n+8\) Divisible by 18? GMAT Data Sufficiency

Question: If n is a non-negative integer, is \(10^n+8\) divisible by 18?

1. n is a prime number.
2. n is even.

1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
2. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
3. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
4. EACH statement ALONE is sufficient.
5. Statements (1) and (2) TOGETHER are NOT sufficient.

“If n is a non-negative integer, is divisible by 18”– is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken f0rom the book "GMAT Quantitative Review". GMAT Quant section consists of a total of 31 questions. GMAT Data Sufficiency questions consist of a problem statement followed by two factual statements. GMAT data sufficiency comprises 15 questions which are two-fifths of the total 31 GMAT quant questions.

Solution and Explanation

Approach Solution 1:

Notice that \(10^n+8\) is divisible by 18 for any positive value of n. In this case, \(10^n+8\) = even + even = even

So it’s divisible by 2.

Also, in this case, the sum of the digits of \(10^n+8\) is 9 so it’s divisible by9.

Since \(10^n+8\) is divisible by both 2 and 9, then it’s divisible by 2*9 = 18 too.

On the other hand, if n = 0 then \(10^n+8\) = 1 + 8 = 9, so in this case \(10^n+8\) is not divisible by 2

(1) n is a prime number. Hence, n is a positive integer.

Sufficient.

(2) n is even. N can be zero as well as any positive even number. Not sufficient

Correct Answer: A

Approach Solution 2:

Non-negative integer means zero, 1,2,3…

Now as (1) says that n is prime, so it cannot be zero. Also (2) says that n is even so it cannot be zero.

So forget zero.

Take n = 1,2,3… in each case, we will get 100 + 8 or 10000 + 8 or 1000000 + 3…

All of these numbers will be divisible by 18 because each of these sums are ending with 8 which is divisible by 2 AND Sum of all digits is 9.. (because there are lot of zeros and 1 and 8 in every sum)

As every number is divisible by 9 and 2… that means it is divisible by 18.

Hence, (1) is sufficient.

So strike out answers BCE.

Now take (2). N is even. In this case also, it is divisible by 18 due to above mentioned logic. Hence strike out A

Answer is D.

Hold on! S2 is not sufficient because if we take n = 0 then we get answer 9 and 9 is not divisible by 18.

Correct Answer: A

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