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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.

Read in the blogs more about the unique learning experience at nubtrek.continue

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`divisibility simplification by factors 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`divisibility simplification by factors 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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