Without dividing, determine if $23$ , $688$ is divisible by $6$ and explain how you know. $\newline$ $23,688 \square \text { divisible by } 6$

Full solution

Q. Without dividing, determine if

23

688

is divisible by

6

and explain how you know.

\newline

23,688 \square \text { divisible by } 6

Check Even Number: To determine if a number is divisible by $6$ , it must be divisible by both $2$ and $3$ . For a number to be divisible by $2$ , it must be even, which means its last digit must be $0$ , $2$ , $4$ , $6$ , or $8$ . Let's check if $23,688$ is even.
Sum of Digits: The last digit of $23,688$ is $8$ , which is an even number. Therefore, $23,688$ is divisible by $2$ .
Check Divisibility by $3$ : Next, to check if a number is divisible by $3$ , the sum of its digits must be divisible by $3$ . Let's add up the digits of $23,688$ : $2 + 3 + 6 + 8 + 8$ .
Check Divisibility by $6$ : The sum of the digits is $2 + 3 + 6 + 8 + 8 = 27$ . Now we need to determine if $27$ is divisible by $3$ .
Check Divisibility by $6$ : The sum of the digits is $2 + 3 + 6 + 8 + 8 = 27$ . Now we need to determine if $27$ is divisible by $3$ .Since $27$ is divisible by $3$ (because $9 \times 3 = 27$ ), the sum of the digits of $23,688$ is divisible by $3$ . Therefore, $23,688$ is divisible by $3$ .
Check Divisibility by $6$ : The sum of the digits is $2 + 3 + 6 + 8 + 8 = 27$ . Now we need to determine if $27$ is divisible by $3$ .Since $27$ is divisible by $3$ (because $9 \times 3 = 27$ ), the sum of the digits of $23,688$ is divisible by $3$ . Therefore, $23,688$ is divisible by $3$ .Since $23,688$ is divisible by both $27$ $1$ and $3$ , we can conclude that $23,688$ is divisible by $27$ $4$ .