Question

Using divisibility tests, determine if the given number is divisible by $2, 3, 4, 5, 6, 7, 8, 9, 10$ or $11$ .

Number	Divisible by
Number	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$	$11$
$504$

Accepted Answer

We know that,

Divisibility by Test

$2$ Last digit is $0$ or even.

$3$ Sum of the digits is divisible by $3$ .

$4$ Last two digits are zeros or the number formed by last two digits is multiple of $4$ .

$5$ Last digit is $0$ or $5$ .

$4$ Number is divisible by $2$ as well as $6$ .

$7$ Difference between twice the units digit and number formed by remaining digits is either $9$ or a multiple of $7$ .

$8$ Number formed by last three digits is a multiple of $8$ .

$9$ Sum of the digits is divisible by $9$ .

$10$ Last digit is $0$ .

$11$ Difference of the sum of digits in odd places and the sum of digits in even places is either $0$ or a multiple of $11$ .

Given number, $504$

It has digit $4$ at its unit place, so the given number is divisible by $2$ .

Sum of the digits $= 5 + 0 + 4 = 9$ is divisible by $3$ , so the given number is divisible by $3$ .

Number formed by last two digits i,e. $04$ is divisible by $4$ , so the given number is divisible by $4$ .

It has digit $4$ at its unit place, so the given number is not divisible by $5$ .

The given number is divisible by $2$ as well as $3$ .

So the given number is divisible by $6$ .

Now,

Twice the units digit $= 2 (4) = 8$

Number formed by remaining digits $= 50$

Difference $= 50 - 8 = 42$ , which is divisible by $7$ .

Therefore, the given number is divisible by $7$ .

Also, number formed by last three digits i,e. $504$ is divisible by $8$ .

So the given number is divisible by $8$ .

Sum of the digits is $9$ which is divisible by $9$ .

So the given number is divisible by $9$ .

As given number does not have unit digit as $0$ , so the given number is not divisible by $10$ .

Now,

Sum of digits at odd places $= 5 + 0 + 4 = 9$

Sum of digits at even places $= 0$

Now, difference of two sums $= 9 - 0 = 9$

So, the given number is not divisible by $11$ .

Hence, $504$ is divisible by $2, 3, 4, 6, 7, 8, 9$ .

Thus, the required details are tabulated as shown below.

Number Divisible by

$2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$ $11$

$504$ Yes Yes Yes No Yes Yes Yes Yes No No

S.N. Chhibber Solutions for Chapter: Playing with Numbers, Exercise 2: Exercise

S.N. Chhibber Mathematics Solutions for Exercise - S.N. Chhibber Solutions for Chapter: Playing with Numbers, Exercise 2: Exercise

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