Question

$67x$ is a three digit number where $x$ is the digit in its units place. Which of the followings could be the value of $x$ if $67x$ is divisible by 7?

• A. 2
• B. 9
• C. 3
• D. 4

Answer 1

Answer: (A), (B)

The correct options are
A 2
B 9

The divisibility rule for 7 states that if a number is divisible by 7 ,then the result obtained by subtracting double the last digit from the remaining number should be 0 or divisible by 7.

$\therefore$ $67x$ is divisible by 7 if $(67-2x)$ is divisible by 7.

$\left(i\right)x=2$
$67-2x=67-2\times 2=67-4=63$ which is divisible by 7.
Hence, 672 is divisible by 7.

$\left(ii\right)x=9$
$67-2x=67-2\times 9=67-18=49$ which is divisible by 7.
Hence, 679 is divisible by 7.

$\left(iii\right)x=3$
$67-2x=67-2\times 3=67-6=61$ which is not divisible by 7.
Hence, 673 is not divisible by 7.

$\left(iv\right)x=4$
$67-2x=67-2\times 4=67-8=59$ which is not divisible by 7.
Hence, 674 is not divisible by 7.