captainPirque wrote:
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?
(A) 9
(B) 16
(C) 23
(D) 35
(E) 47
Hi, could someone please explain this as I am stuck with the same question?
Vegita wrote:
Hi
ScottTargetTestPrepI did 25 + 47 = 72 and the sum 7+2 = 9
I don't understand how your solution worked 2 + 5 + 4 + 7 = 18. Since we have to get the actual value of dividend first and make sure the sum of the dividend is divisible by 9.
To verify whether a number is divisible by 9, we can use the rule that states that the sum of its digits must be divisible by 9. For example, to check if 72 is divisible by 9, we can add its digits (7+2 = 9) and check if the result is divisible by 9. Since 9 is indeed divisible by 9, we can conclude that 72 is also divisible by 9.
This rule can be extended to verify if the sum of any set of integers is divisible by 9. We can simply add all the integers in the set and check if the resulting sum is divisible by 9. Alternatively, we can add the digits of each integer in the set and check if the sum of these digits is divisible by 9. For example, to verify if 40 + 32 is divisible by 9, we can add the digits of each integer (4+0+3+2 = 9) and check if the result is divisible by 9. Since 9 is indeed divisible by 9, we can conclude that 40 + 32 is also divisible by 9.
For the given problem, we can check if \(25*10^n + k*10^{2n}\) is divisible by 9 by either finding the sum of its digits and verifying if it is divisible by 9, or by calculating the sum of the digits of its two components, \(25*10^n\) and \(k*10^{2n}\), separately, and checking if their sum is divisible by 9. Both methods will give the same result.
The sum of the digits of \(25*10^n\) is always equal to 2 + 5 = 7, regardless of the value of n (assuming n is a positive integer). Similarly, the sum of the digits of \(k*10^{2n}\) is equal to the sum of the digits of k, again regardless of the value of n (assuming n is a positive integer).
Therefore, to find whether \(25*10^n + k*10^{2n}\) is divisible by 9, we can add 7 to the sum of the digits of k and check if the resulting sum is divisible by 9. Among the answer choices, only 47 (option E) satisfies this condition, where the sum of its digits (4+7) added to 7 gives a total sum that of 18, which is divisible by 9.
Hope it's clear.