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A positive integer is divisible by 9 if and only if the sum of its digits is divisible by 9. If a and b are positive integers with b = 2 a, for which of the following values of k is 17 \times 1 0^{a} + k \times 1 0^{b} divisible by 9?

Explanation	Calculations	Help
$We apply $\textbf${}, which states that a number is divisible by 9 if and only if the sum of its digits is a multiple of 9.$	$A number is divisible by 9 ⟺ sum of its digits is a multiple of 9$	Theory & Tactics Method Card DIV3-C Click to view full details
$Appending zeros does not change the nonzero digits. We find the sum of the digits of 17 \times 1 0^{a} .$	$sum of digits (17 \times 1 0^{a}) = 1 + 7 = 8$
$Similarly, appending zeros does not change the digits of k . We find the sum of the digits of k \times 1 0^{b} .$	$sum of digits (k \times 1 0^{b}) = sum of digits (k)$
$We add these sums to get the total sum of digits of the entire expression.$	$sum of digits (17 \times 1 0^{a} + k \times 1 0^{b}) = 8 + sum of digits (k)$
$We test each answer choice to see which total is a multiple of 9.$	$8 + sum of digits (16) = 8 + 7 = 15 8 + sum of digits (23) = 8 + 5 = 13 8 + sum of digits (30) = 8 + 3 = 11 8 + sum of digits (37) = 8 + 10 = 18 8 + sum of digits (44) = 8 + 8 = 16$
$Only when k = 37 is the digit sum 18, which is a multiple of 9.$	$Answer = 37$

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D

A positive integer is divisible by 9 if and only if the sum of its digits is divisible by 9. If a and b are positive integers with b = 2 a, for which of the following values of k is 17 \times 1 0^{a} + k \times 1 0^{b} divisible by 9?

Explanation	Calculations	Help
$We apply $\textbf${}, which states that a number is divisible by 9 if and only if the sum of its digits is a multiple of 9.$	$A number is divisible by 9 ⟺ sum of its digits is a multiple of 9$	Theory & Tactics Method Card DIV3-C Click to view full details
$Appending zeros does not change the nonzero digits. We find the sum of the digits of 17 \times 1 0^{a} .$	$sum of digits (17 \times 1 0^{a}) = 1 + 7 = 8$
$Similarly, appending zeros does not change the digits of k . We find the sum of the digits of k \times 1 0^{b} .$	$sum of digits (k \times 1 0^{b}) = sum of digits (k)$
$We add these sums to get the total sum of digits of the entire expression.$	$sum of digits (17 \times 1 0^{a} + k \times 1 0^{b}) = 8 + sum of digits (k)$
$We test each answer choice to see which total is a multiple of 9.$	$8 + sum of digits (16) = 8 + 7 = 15 8 + sum of digits (23) = 8 + 5 = 13 8 + sum of digits (30) = 8 + 3 = 11 8 + sum of digits (37) = 8 + 10 = 18 8 + sum of digits (44) = 8 + 8 = 16$
$Only when k = 37 is the digit sum 18, which is a multiple of 9.$	$Answer = 37$

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D