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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.

This introduces prime and composite numbers with simple examples.

click on the content to continue..

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

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

• Pronunciation : Say the answer once

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

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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