Question

Considering the remainder of the place values, check which of following numbers are divisible by seven.

• A. 108402
• B. 865578
• C. 398625
• D. 179562

Answer 1

Answer: (A), (B)

The correct options are
A 108402
B 865578

In this method we take remainders by dividing the place values, with the given number which is seven in our case.

If we divide the place values of a number by 7, we get the remainders as
$1000÷7$ (Remainder 6. This can be taken as $6-7=-1$)
$100÷7$ (Remainder 2)
$10÷7$ (Remainder 3)
$1÷7$ (Remainder 1)

Place Value	${10}^5$	${10}^4$	${10}^3$	${10}^2$	${10}^1$	${10}^0$
Remainders divide by 7	$-2$	$-3$	$-1$	$2$	$3$	$1$

Now to check whether 108402 is divisible by 7 or not

Digits	$1$	$0$	$8$	$4$	$0$	$2$
Place Value	${10}^5$	${10}^4$	${10}^3$	${10}^2$	${10}^1$	${10}^0$
Remainders divide by 7	$1\times (-2)$	$0\times (-3)$	$8\times (-1)$	$4\times 2$	$0\times 3$	$2\times 1$

Sum of product of face values and remainders of the place values is -2 + 0 - 8 + 8 + 0 + 2 = 0
Hence, 108402 is divisible by 7.

To check whether 865578 is divisible by 7 or not

Digits	$8$	$6$	$5$	$5$	$7$	$8$
Place Value	${10}^5$	${10}^4$	${10}^3$	${10}^2$	${10}^1$	${10}^0$
Remainders divide by 7	$8\times (-2)$	$6\times (-3)$	$5\times (-1)$	$5\times 2$	$7\times 3$	$8\times 1$

Sum of product of face values and remainders of the place values = -16 - 18 - 5 + 10 + 21 + 8 = 0 (divisible by 7)
Thus, 865578 is divisible by 7.

To check whether 398625 is divisible by 7 or not

Digits	$3$	$9$	$8$	$6$	$2$	$5$
Place Value	${10}^5$	${10}^4$	${10}^3$	${10}^2$	${10}^1$	${10}^0$
Remainders divide by 7	$3\times (-2)$	$9\times (-3)$	$8\times (-1)$	$6\times 2$	$2\times 3$	$5\times 1$

Sum of product of face values and remainders of the place values is -18 (not divisible by 7)
So, 398625 is not divisible by 7.

To check whether 179562 is divisible by 7 or not

Digits	$1$	$7$	$9$	$5$	$6$	$2$
Place Value	${10}^5$	${10}^4$	${10}^3$	${10}^2$	${10}^1$	${10}^0$
Remainders divide by 7	$1\times (-2)$	$7\times (-3)$	$9\times (-1)$	$5\times 2$	$6\times 3$	$2\times 1$

Sum of product of face values and remainders of the place values is -2 (not divisible by 7)
Therefore, 179562 is not divisible by 7.