Fitzgerald Wagner
07/09/2023 · Elementary School
Determine whether the following divisibility statement is true of false. Justify your solution. \[ 9 \mid 1017 \] False, because the sum of the digits of 1017 is not divisible by 9. True, because the sum of the digits of 1017 is divisible by 9. False, because the last two digits do not form a number that is divisible by 9. True, because the last two digits form a number that is divisible by 9.
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Step-by-step Solution
To determine whether \( 9 \mid 1017 \) (which means "9 divides 1017"), we can use the rule of divisibility for 9. According to this rule, a number is divisible by 9 if the sum of its digits is divisible by 9.
Let's calculate the sum of the digits of 1017:
- The digits of 1017 are 1, 0, 1, and 7.
- The sum of these digits is:
\[
1 + 0 + 1 + 7 = 9
\]
Now, we check if this sum (9) is divisible by 9:
\[
9 \div 9 = 1
\]
Since 1 is an integer, 9 is indeed divisible by 9.
Thus, the statement \( 9 \mid 1017 \) is **true**.
The correct justification is:
**True, because the sum of the digits of 1017 is divisible by 9.**
Quick Answer
The statement \( 9 \mid 1017 \) is true because the sum of the digits of 1017 is divisible by 9.
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