UNDERSTANDING PRIME NUMBERS THE SHORT WAY

Using the logic that we have to look at only the prime numbers below the square root in order to check whether a number is prime, we can actually cut short the time for finding whether a number is prime drastically.

geralt / Pixabay

Before I start to explain this, you should perhaps realise that in an examination like the CAT, whenever you would need to be checking for whether a number is prime or not, you would typically be checking 2 digit or maximum 3 digit numbers in the range of 100 to 200.

Also, one would never really need to check with the prime number 5, because divisibility by 5 would automatically be visible and thus, there is no danger of anyone ever declaring a number like 35 to be prime. Hence, in the list of prime numbers below the square root we would never include 5 as a number to check with.

Checking Whether a Number is Prime (For Numbers below 49)
The only number you would need to check for divisibility with is the number 3. Thus, 47 is prime because it is not divisible by 3.
Checking Whether a Number is Prime (For Numbers above 49 and below 121)

Naturally you would need to check this with 3 and 7. But if you remember that 77, 91 and 119 are not prime, you would be able to spot the prime numbers below 121 by just checking for divisibility with the number 3.

Why? Well, the odd numbers between 49 and 121 which are divisible by 7 are 63, 77, 91, 105 and 119. Out of these perhaps 91 and 119 are the only numbers that you can mistakenly declare as prime. 77 and 105 are so obviously not-prime that you would never be in danger of declaring them prime.

Thus, for numbers between 49 and 121 you can find whether a number is prime or not by just dividing by 3 and checking for its divisibility. For example:

61, is prime because it is not divisible by 3 and it is neither 91 nor 119.

Checking Whether a Number is Prime (For Numbers above 121 and below 169)

geralt / Pixabay

Naturally you would need to check this with 3, 7 and 11. But if you remember that 133,143 and 161 are not prime, you would be able to spot the prime numbers between 121 to 169 by just checking for divisibility with the number 3. Why? The same logic as explained above. The odd numbers between 121 and 169 which are divisible by either 7 or 11 are 133,143,147,161 and 165. Out of these 133,143 and 161 are the only numbers that you can mistakenly declare as prime if you do not check for 7 or 11. The number 147 would be found to be not prime when you check its divisibility by 3 while the number 165 you would never need to check for, for obvious reasons.

Thus, for numbers between 121 and 169 you can find whether a number is prime or not by just dividing by 3 and checking for its divisibility.
For example:

149, is prime because it is not divisible by 3 and it is neither 133,143 nor 161.
Thus, we have been able to go all the way till 169 with just checking for divisibility with the number 3.
This logic can be represented on the number line as follows:
Integers A set which consists of natural numbers, negative integers (– 1, –2, –3…– n…) and zero is known as the set of integers. The numbers belonging to this set are known as integers.

The Number System covers 2-3 questions in Quant Section of competitive exams like TOEFL. It is one of the easiest quant areas to prepare for, but the way CAT questions are asked from this section varies a lot given the huge fundamental concepts that it covers.Prime numbers also form a part of this section.


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