Server Error

Server Not Reachable.

This may be due to your internet connection or the nubtrek server is offline.

Thought-Process to Discover Knowledge

Welcome to nubtrek.

Books and other education websites provide "matter-of-fact" knowledge. Instead, nubtrek provides a thought-process to discover knowledge.

In each of the topic, the outline of the thought-process, for that topic, is provided for learners and educators.

Read in the blogs more about the unique learning experience at nubtrek.
mathsDivisibility in Whole NumbersSimplification of Divisibility Tests: 8, 12, 15

### Divisibility: Simplification by Factors

In this page, Simplification of divisibility test by factors is explained. This is used to develop other divisibility tests for specific numbers.

click on the content to continue..

Given a number 2321. Is it divisible by 58?

Check this by long division method or by simplification by subtraction.

• Yes
• No
• No
• Can't say

The answer is "No". 2321-58xx4 = 1. So, the number is not divisible by 58

Consider the divisibility of dividend 2321 by 58.
An observation is that if a number is multiple of divisor 58, then the number can be written as 58 xx n. The same can be written as 2xx29xxn, because 58=2xx29.

What does this imply?

• the dividend is a multiple of 2 and 29
• the dividend is a multiple of 2 and 29
• this does not imply anything

The answer is "the dividend is a multiple of 2 and 29". Since 2321 is an odd number, it is not divisible by 2 and so it is not divisible by 58.

Consider the divisibility of 922 by 58.
It is observed that both are even numbers. So 922 = 2 xx 461 and 58=2xx29.

What does this imply?

• the factor 461 should be a multiple of 29 for the divisibility test to pass
• the factor 461 should be a multiple of 29 for the divisibility test to pass
• this does not mean anything

The answer is "the factor 461 should be a multiple of 29 for the divisibility test to pass". That is divisibility test of 922 by 58 is now changed to divisibility test of 461 by 29. The common factor 2 is removed to simplify.

To check the divisibility of a dividend by a composite divisor, the factors of the divisors can help in simplifying the divisibility.

•  If the dividend is not divisible by a factor of divisor, then the dividend is not divisible by the composite divisor.
eg: divisibility of 2321 by 58. 58 = 2xx 29. The dividend 2321 is not divisible by a factor of divisor 2, so 2321 is not divisible by 58.

•  If the dividend is divisible by a factor, then both the dividend and divisor can be divided by the factor and the test is done on the results.
eg: Divisibility of 922 by 58. Both has common factor 2 and so it is simplified as divisibility of 461 by 29. This is named as Simplification by Factors.

Simplification by Factors: To simplify divisibility test of dividend by a divisor,

•  any common factors can be removed from both the numbers and divisibility test can be performed on simplified dividend by the simplified divisor.

•  the dividend has to have all the factors of the divisor to pass the divisibility test.

Check divisibility of 2016 by 264 The figure shows a simplified procedure by which simplification by factors can be performed.

• Divisible
• Not Divisible
• Not Divisible

Check divisibility of 3120 by 156 The figure shows the simplified procedure by which simplification by factors is performed.