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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.continue

This introduces prime and composite numbers with simple examples.



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Which of the following describes the numbers `7` and `8`?

  • `7` is not divisible by any numbers between `2` and `6` (both inclusive)
  • `8` is divisible by some numbers between `2` and `7` (that is, divisible by `2` and `4`)
  • both the above
  • both the above

The answer is "both the above"

Some numbers are not divisible by any numbers between `2` and one less than the number.
eg: `5` is not divisible by any of `2`, `3`, `4`.
And `13` is not divisible by `2`, `3`, `4`, `5`, `6`, `7`, `8`, `9`, `10`, `11`, and `12`.

Such numbers are called "prime numbers". They have the distinct characteristic of not divisible by any number other than `1` and itself.
Examples of Prime numbers: `2`, `3`, `5`, `7`, `13`, `17`, `cdots`

Some numbers are divisible by at-least one number between `2` and one less than the number.
eg: `6` is divisible by `2` and `3`.
And `63` is divisible by `3`, `7`, `9`, and `21`.

Such numbers are called "composite numbers". They can be equivalently given as a product of other numbers.
eg: `6 = 2xx3` and `63 = 3xx21` or `63 = 7xx9`.

Which of the following is a meaning for the word "composite"?

  • made up of several parts or elements
  • made up of several parts or elements
  • expressing dissatisfaction

The answer is "made up of several parts or elements".

What is the term used to refer to the numbers that can be represented as product of other numbers?

  • Pronunciation : Say the answer once
    Spelling: Write the answer once

The answer is "composite numbers".

Which of the following is a meaning for the word "prime"?

  • leader of self-configuring alien robotic lifeforms
  • having distinct or important properties
  • having distinct or important properties

The answer is "having distinct or important properties".

What is the term used to refer to the numbers that are divisible by only `1` and itself?

  • Pronunciation : Say the answer once
    Spelling: Write the answer once

The answer is "prime numbers".

Composite Numbers : Numbers that are divisible by at-least a number other than `1` and the number itself.

Prime Numbers : Numbers that are divisible by only `1` and the number itself.

We need to find if `59` is a prime or composite number. Which of the following helps to conclude that?

  • Check for divisibility by numbers from `2` to `58`
  • Check for divisibility by numbers from `2` to `7`
  • Check for divisibility by numbers from `2` to `7`
  • any one of the above will work

The answer is "Any one of the above".

By the definition of prime numbers, the number can be checked for divisibility by numbers from `2` to `58`.

But, it is sufficient to check for divisibility by numbers from `2` to `7`. The number `7` is chosen because the given number `59` lies between the perfect squares `7xx7=49` and `8xx8=64`

To check if a number is prime, the divisibility test is done for numbers from `2` to the highest number which, when squared, is lesser than the given number.

eg: `93` is to be checked if it is prime. The check can be from `2` to `9`, as `9xx9=81` which is less than the given number `93`. But `10xx10=100`, which is greater than the given number `93`.
It is noted that `31` is a factor, but there is a smaller factor less than `7`. That is, `3` is a factor `93=3xx31`. For any factor greater than the limit given, there is a smaller factor within the limit.

Is `39` a prime number?

  • Yes
  • No
  • No

The answer is "No". `39` is divisible by `3`.

is `59` a prime?

  • yes
  • yes
  • no

The answer is "yes". Checking from `2` to `7` it is decided that `59` is a prime number.

                            
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