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Thought-Process to Discover Knowledge

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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.

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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.

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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

The answer is "Not divisible"

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

• Divisible
• Not Divisible
• Not Divisible

The answer is "Divisible".

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