maths > wholedivisors

Introduction to Divisibility Tests

what you'll learn...

overview

•  provides basics required to understand what is a divisibility test.

•  analyses developing divisibility test procedure for numbers given as product of multiple numbers. This forms the foundation to developing other divisibility tests for specific numbers.

•  develops divisibility test procedure for numbers given as sum of multiple numbers. This forms the foundation to developing other divisibility tests for specific numbers.

Understanding Divisibility

Note the difference

•  Result of 5÷2$5 \div 2$ is, quotient 2$2$ and remainder 1$1$

•  Result of 6÷2$6 \div 2$ is, quotient 3$3$ and remainder 0$0$.

For a given divisor, the numbers can be grouped as

•  numbers for which remainder is $0$ and

•  numbers for which remainder is not $0$.

A number is called "divisible" by a divisor, if the remainder is $0$.

Divisibility : A number (dividend) is divisible by a divisor number if the remainder is $0$.

Divisibility Test : A procedure, to check if a given number is divisible by a divisor or not divisible by a divisor, is called divisibility test of the divisor.

Divisibility of product of numbers

The number $21$ can be equivalently given as $3 \times 7$. That is $3 \times 7$ is a number, which is given as an expression.

In this expression form, the number $3 \times 7$ is readily understood to be divisible by $3$, as the number is a multiple of $3$. No need to multiply and then check if the product is divisible by $3$. The divisibility is evident from the given expression.

The number $6 \times 7$ is divisible by $3$, as a factor $6$ is divisible by $3$. No need to multiply and then check if the product is divisible by $3$. The divisibility is evident from the given expression.

If a multiple is given as a product of two or more numbers (eg: "multiplicand $\times$ multiplier"), then if one of them is divisible by a divisor, then the product is divisible by the divisor.

eg: Consider the number $6 \times 25$. The multiplicand $6$ is divisible by $3$, so the product $6 \times 25$ is divisible by $3$.

Divisibility test of a Product: A number is given as product of two numbers : multiplicand $\times$ multiplier.

If multiplicand or multiplier is divisible, then the product is divisible.

Divisibility of sum of numbers

The number $15$ can be equivalently given as $9 + 6$. That is $9 + 6$ is a number, which is given as an expression.

Consider the number $9 + 6$. It is known that addends $9$ and $6$ are divisible by $3$. Since the addends are divisible, the sum $9 + 6$ is divisible by $3$.

Consider the number $9 - 6$. It is known that minuend $9$ and subtrahend $6$ are divisible by $3$. So, the difference $9 - 6$ is divisible by $3$.

If a number is given as sum of two addends, and one of the addends is divisible, then the divisibility of the sum is decided by the other addend.
eg: Is $6 + 8$ divisible by $3$? Since $6$ is divisible by $3$, the divisibility of $6 + 8$ is decided by divisibility of $8$ by $3$.

If a number is given as difference of two numbers, and one of the numbers is divisible, then the divisibility of the difference is decided by the other number.

eg: Is $15 - 6$ divisible by $3$? Since $6$ is divisible by $3$, the divisibility of $15 - 6$ is decided by divisibility of $15$ by $3$.

eg: Is $3000 - 23$ divisible by $3$? Since $3000$ is divisible by $3$, the divisibility of $3000 - 23$ is decided by divisibility of $23$ by $3$.

We studied that divisibility test of sum or difference can be simplified into divisibility test on addend or minuend or subtrahend. This property is further developed to simplify divisibility test as follows.

To simplify divisibility test of a number, a multiple of divisor can be added or subtracted. The divisibility test can be performed on the result.

eg: To find if $892$ is divisible by $83$, subtract $830$ from $892$, and the result $62$ is not divisible by $83$ and so $892$ is not divisible by $83$.

Property of Divisibility of sum or difference: If a number is given as sum of two numbers left addend + right addend. Then if left addend is divisible, then divisibility of the sum is decided by the divisibility of the right addend.

If a number is given as difference of two numbers (minuend - subtrahend) and one of the numbers is divisible, then the divisibility of the difference is decided by the divisibility of the other number.

Simplification of Divisibility by Addition or Subtraction : Divisibility test of a divisor on a number can be simplified by adding or subtracting a multiple of the divisor from the number.

Consider the number given as $34 \times 21 + 3$. Is the number divisible by $34$?
The number is given as sum of two numbers. One addend is divisible by $34$. The divisibility is decided by the second addend $3$. The second addend is not divisible by $34$ and so the number $34 \times 21 + 3$ is not divisible by $34$.

Is $22 \times 50 - 22 \times 23$ divisible by $22$?
The answer is "Yes". Both minuend and subtrahend are multiples of $22$.

summary

Divisibility Test : A procedure, to check if a given number is divisible by a divisor or not divisible by a divisor, is called divisibility test of the divisor.

Divisibility test of a Product: A number is given as product of two numbers : multiplicand $\times$ multiplier.

If multiplicand or multiplier is divisible, then the product is divisible.
eg: $6 \times 25$ is divisible by $3$ as $6$ is divisible by $3$

Property of Divisibility of sum or difference: If a number is given as sum of two numbers left addend + right addend. Then if left addend is divisible, then divisibility of the sum is decided by the divisibility of the right addend.

If a number is given as difference of two numbers (minuend - subtrahend) and one of the numbers is divisible, then the divisibility of the difference is decided by the divisibility of the other number.
eg: the divisibility by $3$ of $84 - 6$, is equivalently divisibility by $3$ of $84$, as $6$ is divisible by $3$.

Simplification of Divisibility by Subtraction : Divisibility test of a divisor on a number can be simplified by adding or subtracting a multiple of the divisor from the number.
eg: the divisibility by $3$ of $84$, is equivalently divisibility by $3$ of $84 - 6$, as $6 = 2 \times 3$

Outline

The outline of material to learn "Divisibility in Whole Numbers" is as follows.

→   Classification as odd, even, prime, and composite

→   Factors, Multiples, Prime factorization

→   Highest Common Factor

→   Lowest Common Multiple

→   Introduction to divisibility tests

→   Simple Divisibility Tests

→   Simplification of Divisibility Tests

→   Simplification in Digits for Divisibility Tests