Name: Test for Divisibility by 9
Uploaded: 2019-07-19
Duration: 5 min 33 s
Description: The sum of digits of any given number will help you determine if that particular number is divisible by 9 or not! Sounds interesting? Watch this video to und...

Question

Search and write the rules for divisibility by $15, 18, 45$ and $66$ with at least $2$ examples of each.

Accepted Answer

We know that, if a number is divisible by another number, then it is divisible by each of the factors of that number.

We know that $15 = 3 \times 5$

Hence, a number is divisible by $15$ , if it is divisible by both $3$ and $5$ .

A number is divisible by $3$ , if the sum of its digits is divisible by $3$ .

A number is divisible by $5$ , if its units digit is either $0$ or $5$ .

Example: $30$ and $45$ .

Sum of digits of $30$ is $3 + 0 = 3$ , which is divisible by $3$ . Similarly, sum of digits of $45$ is $4 + 5 = 9$ , which is divisible by $3$ .

We know that $18 = 2 \times 9$

Hence, a number is divisible by $18$ , if it is divisible by both $2$ and $9$ .

A number is divisible by $2$ , if its unit digit is an even number.

A number is divisible by $9$ , if the sum of its digits is divisible by $9$ .

Example: $36$ and $54$

Unit digit of $36$ and $54$ are even numbers. Hence, they are divisible by $2$ .

Sum of digits of $36$ is $3 + 6 = 9$ , which is divisible by $9$ . Similarly, sum of digits of $45$ is $4 + 5 = 9$ , which is divisible by $9$ .

We know that $45 = 5 \times 9$

Hence, a number is divisible by $45$ , if it is divisible by both $5$ and $9$ .

A number is divisible by $5$ , if its units digit is either $0$ or $5$ .

A number is divisible by $9$ , if the sum of its digits is divisible by $9$ .

Example: $135$ and $360$ .

Unit digits of $135$ and $360$ are $5$ and $0$ respectively.

Sum of digits of $135$ is $1 + 3 + 5 = 9$ , which is divisible by $9$ . Similarly, sum of the digits of $360$ is $3 + 6 + 0 = 9$ , which is divisible by $9$ .

We know that $66 = 2 \times 3 \times 11$

Hence, a number is divisible by $66$ , if it is divisible by $2, 3$ and $11$ .

A number is divisible by $2$ , if its unit digit is an even number.

A number is divisible by $3$ , if the sum of its digits is divisible by $3$ .

A number is divisible by $11$ , if the difference between the sum of the digits at odd places and the sum of the digits at even places is either zero or a multiple of $11$ .

Example: $330$ and $528$

Unit digit of $330$ and $528$ are even numbers. Hence, they are divisible by $2$ .

Sum of digits of $330$ is $3 + 3 + 0 = 6$ , which is divisible by $3$ . Similarly, sum of digits of $528$ is $5 + 2 + 8 = 15$ , which is divisible by $3$ .

Difference of sums $=$ Sum of digits at odd places $-$ Sum of digits at even places

For $330$ , we have $(3 + 0) - 3 = 3 - 3 = 0$ , which is zero.

For $528$ , we have $(5 + 8) - 2 = 13 - 2 = 11$ , which is a multiple of $11$ .

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