A Positive Integer Is Divisible by 9 If And Only If The Sum of Its Digits is Divisible By 9 GMAT Problem Solving

Question: A positive integer is divisible by 9 if and only if the sum of its digits is divisible by 9. If n is a positive integer, for which of the following values of k is 25∗\(10^n+k∗10^{2n}\) divisible by 9?

1. 9
2. 16
3. 23
4. 35
5. 47

“A positive integer is divisible by 9 if and only if the sum of its digits is divisible by 9.”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Quantitative Review”. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

Solution and Explanation:

Approach Solution 1:
Given:

• A positive integer is divisible by 9 if and only if the sum of its digits is divisible by 9.
• n is a positive integer

Find Out:

• We need to determine for which value of k 25*10^n + k*10^2n is divisible by 9.

We see that 10^n and 10^2n will always have a digit of 1 and then zeros.
Hence, excluding k, the sum of the digits in our expression is
2 + 5 = 7 (since (25)(10^n) has a 2, 5, and zeros).
We need to determine, of our answer choices, which when added to 7 will produce a sum that is divisible by 9.
Scanning our answer choices, we see that 47 is the correct answer.
2 + 5 + 4 + 7 = 18, which is divisible by 9.

Correct Answer: E

Approach Solution 2:
Given:

• A positive integer is divisible by 9 if and only if the sum of its digits is divisible by 9.
• n is a positive integer

Find Out:

• We need to determine for which value of k 25*10^n + k*10^2n is divisible by 9.

Solution:
25*10^n + k*10^2n = (25 * 10^n) + (k* 10^n * 10^n)
We should be able to see that 10^2n breaks into 10^n * 10^n
We can also work the exponents roots in algebra
Next what we will do is 10^n * [ 25 +(k * 10^n) ]
And here the logic/critical thinking begins
Now, we will position the logic behind this.
a) The only thing that the very first 10^n does to the number within the square brackets is simply padding it with zero at the end. (as other users suggested before).
Hence, it doesn't play any role to the divisibility hence can be fully ignored (N.B. n>0).
b) now 25 + (k * 10^n) simply is K * mul(10) + 25
Therefore we care only about the sum of the digits K, 2 and 5.
By using "back-solving" we can find E

Correct Answer: E

Approach Solution 3:

Given:

If and only if the sum of a positive integer's digits is 9-divisible, then number is said to be positive integer n.

Figure out:

Find the value of k for which 25*10n + k*102n is divisible by 9.
We can see that numbers 10 and 102 will always be followed by zeros.
Since (25)(10n) includes a 2, 5, and zeros, the total of the digits in our formula, omitting k, is 2 + 5 = 7.
Which of our possible answers, when added to 7, will result in a sum that can be divided by nine?
Examining our potential responses, we discover that 47 is the right response.
2 + 5 + 4 + 7 = 18, a number that may be divided by 9

Correct Answer: E

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