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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 m is a positive integer, for which of the following values of k is 314 \times 1 0^{m} + k \times 1 0^{3 m + 1} divisible by 9?

Explanation	Calculations	Help
$We apply the divisibility rule for 9, which states that a number is divisible by 9 exactly when the sum of its digits is a multiple of 9.$	$A number is divisible by 9 ⟺ sum of its digits is a multiple of 9$	Theory & Tactics Method Card DIV3-C Click to view full details
$We find the sum of the digits of the term 314 \times 1 0^{m} . Appending zeros does not change the nonzero digits.$	$sum of digits (314 \times 1 0^{m}) = 3 + 1 + 4 = 8$
$We find the sum of the digits of the term k \times 1 0^{3 m + 1} . Appending zeros does not change the digit sum of k .$	$sum of digits (k \times 1 0^{3 m + 1}) = sum of digits (k)$
$We add these digit sums to get the total digit sum of the expression.$	$sum of digits (314 \times 1 0^{m} + k \times 1 0^{3 m + 1}) = 8 + sum of digits (k)$
$For divisibility by 9, the total must be a multiple of 9. We test each choice by computing 8 + sum of digits (k) .$	$8 + sum of digits (15) = 8 + (1 + 5) = 14 8 + sum of digits (19) = 8 + (1 + 9) = 18 8 + sum of digits (23) = 8 + (2 + 3) = 13 8 + sum of digits (30) = 8 + (3 + 0) = 11 8 + sum of digits (39) = 8 + (3 + 9) = 20$
$We see that only for the choice 19 the total digit sum is a multiple of 9, so we select 19.$	$Answer = 19$

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B

A positive integer is divisible by 9 if and only if the sum of its digits is divisible by 9. If m is a positive integer, for which of the following values of k is 314 \times 1 0^{m} + k \times 1 0^{3 m + 1} divisible by 9?

Explanation	Calculations	Help
$We apply the divisibility rule for 9, which states that a number is divisible by 9 exactly when the sum of its digits is a multiple of 9.$	$A number is divisible by 9 ⟺ sum of its digits is a multiple of 9$	Theory & Tactics Method Card DIV3-C Click to view full details
$We find the sum of the digits of the term 314 \times 1 0^{m} . Appending zeros does not change the nonzero digits.$	$sum of digits (314 \times 1 0^{m}) = 3 + 1 + 4 = 8$
$We find the sum of the digits of the term k \times 1 0^{3 m + 1} . Appending zeros does not change the digit sum of k .$	$sum of digits (k \times 1 0^{3 m + 1}) = sum of digits (k)$
$We add these digit sums to get the total digit sum of the expression.$	$sum of digits (314 \times 1 0^{m} + k \times 1 0^{3 m + 1}) = 8 + sum of digits (k)$
$For divisibility by 9, the total must be a multiple of 9. We test each choice by computing 8 + sum of digits (k) .$	$8 + sum of digits (15) = 8 + (1 + 5) = 14 8 + sum of digits (19) = 8 + (1 + 9) = 18 8 + sum of digits (23) = 8 + (2 + 3) = 13 8 + sum of digits (30) = 8 + (3 + 0) = 11 8 + sum of digits (39) = 8 + (3 + 9) = 20$
$We see that only for the choice 19 the total digit sum is a multiple of 9, so we select 19.$	$Answer = 19$

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B