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

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 10^2 n divisible by 9 ?

GMAT Panda · Accepted Answer

47 We apply the divisibility rule for 9, which states that a number is divisible by 9 exactly when the sum of its digits is a multiple of 9. A number is divisible by 9 sum of its digits is a multiple of 9 We find the digit sum of the term 25 10^n. Appending zeros does not change the nonzero digits. sum of digits(25 10^n) = 2 + 5 = 7 We find the digit sum of the term k 10^2n. Appending zeros does not change the digit sum of k. sum of digits(k 10^2n) = sum of digits(k) We add these digit sums to get the total digit sum of the entire expression. sum of digits(25 10^n + k 10^2n) = 7 + sum of digits(k) For divisibility by 9, the total must be a multiple of 9. We test each choice by computing 7 + sum of digits(k). 7 + sum of digits(9) = 7 + 9 = 16 7 + sum of digits(16) = 7 + (1 + 6) = 14 7 + sum of digits(23) = 7 + (2 + 3) = 12 7 + sum of digits(35) = 7 + (3 + 5) = 15 7 + sum of digits(47) = 7 + (4 + 7) = 18 We see that only for the choice 47 the total digit sum is a multiple of 9, so we select 47. Answer = 47

Explanation	Calculations	Help
$We apply the divisibility rule for 9, which states that a number is divisible by 9 exactly when the sum of its digits is a multiple of 9.$	$A number is divisible by 9 ⟺ sum of its digits is a multiple of 9$
$We find the digit sum of the term 25 \times 1 0^{n} . Appending zeros does not change the nonzero digits.$	$sum of digits (25 \times 1 0^{n}) = 2 + 5 = 7$
$We find the digit sum of the term k \times 1 0^{2 n} . Appending zeros does not change the digit sum of k .$	$sum of digits (k \times 1 0^{2 n}) = sum of digits (k)$
$We add these digit sums to get the total digit sum of the entire expression.$	$sum of digits (25 \times 1 0^{n} + k \times 1 0^{2 n}) = 7 + sum of digits (k)$
$For divisibility by 9, the total must be a multiple of 9. We test each choice by computing 7 + sum of digits (k) .$	$7 + sum of digits (9) = 7 + 9 = 16 7 + sum of digits (16) = 7 + (1 + 6) = 14 7 + sum of digits (23) = 7 + (2 + 3) = 12 7 + sum of digits (35) = 7 + (3 + 5) = 15 7 + sum of digits (47) = 7 + (4 + 7) = 18$
$We see that only for the choice 47 the total digit sum is a multiple of 9, so we select 47.$	$Answer = 47$

Question

Options:

Steps

Final Answer

Question

Options:

Steps

Final Answer

Related questions