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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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This page provides a brief overview of finding highest common factor using division method.



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Consider two numbers with HCF `n`. The numbers can be given in the form `nxxp` and `nxxq`.

For example, HCF `36` and `48` is `12`. And `36=12xx3` and `48=12xx4`. In this example `n=12`, `p=3` and `q=4`.

The `p` and `q` are derived from non-common prime factors of the two numbers.

Let us consider that `nxxq` is the larger than `nxxp`.

To simplify finding HCF, the larger number is subtracted by a multiple of the smaller number. That is, `nxxq - mxxnxxp` is the difference, where `m` is randomly chosen.

What is the HCF of the two numbers `nxxp` and `nxxq-mxxnxxq`?

  • same as the HCF of `nxxp` and `nxxq`
  • the HCF cannot be found with modified numbers
  • the HCF cannot be found with modified numbers

The answer is "same as the HCF of `nxxp` and `nxxq`".

It was learned that HCF of two numbers `nxxp` and `nxxq` is same as the HCF of `nxxp` and `nxxq - mxxnxxp` for any `m`. Using this property, a simplified procedure is devised.finding HCF using division method The simplified procedure is given in the figure.

 •  The numbers are placed in long division form

 •  A multiple of smaller number is subtracted from the larger number (`420-280`).

 •  Now, the difference and the smaller number are the pair for which the HCF is to be found. repeat the procedure in the next step. The step in which the difference is `0`, the smaller number in that step is the HCF.

Division Method to find HCF : To find HCF of two large numbers :

 •  The numbers are placed in long division form

 •  A multiple of smaller number is subtracted from the larger number.

 •  Now, the difference and the smaller number are the pair for which the HCF is to be found. repeat the procedure in the next step. The step in which the difference is `0`, the smaller number in that step is the HCF.finding HCF using division method An example is illustrated in the figure.

Find HCF of `336` and `72`.

  • `24`
  • `24`
  • `12`

The answer is "`24`".
hcf

                            
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