Question

The difference between the sum of prime numbers from 21 to 30 and the sum of the composite numbers from 21 to 30 is divisible by which of the following numbers:-
(A) 3 (B) 7 (C) 151 (D) 17

a)Divisible by (A) and (B)
b)Divisible by only (B)
c)Divisible by only (C)
d)Divisible by only (D)

Answer 1

Hint:- For solving this question, we will have to follow the following steps.
Firstly, we shall write the prime numbers from 21 to 30 and add them. Then, we shall write the composite numbers from 21 to 30 and add them also.
Then, we shall find the difference between their sums by subtracting the sum of the two numbers.
Then, we shall check for the divisibility of 3, 7, 151 and 17.
And hence, we shall obtain the answer.

Complete step-by-step answer:
Let us now solve this question.
Prime numbers from 21 to 30 = 23 and 29
Sum of the prime numbers from 21 to 30 = 23 + 29 = 52
Composite numbers from 21 to 30 = 21, 22, 24, 25, 26, 27, 28 and 30
Sum of the composite numbers from 21 to 30 = 21 + 22 + 24 + 25 + 26 + 27 + 28 + 30 = 203

Difference between the sum of prime numbers and composite numbers from 21 to 30 = 203 – 52
                                                                                                                                                            = 151
So, the difference between the sum of prime numbers and composite numbers from 21 to 30 is 151.
Now, as we know that 151 is a prime number.
Prime numbers are the numbers with just 2 factors, first one is 1 and the other is the number itself.
So, as 151 is a prime number, the only two factors of 151 are 1 and 151.
Now, the only option that matches with the above explanation is option (C) 151.
So, the correct option for this question is (c) Divisible by only (C).

Note:-Let us know about the divisibility tests of a few numbers.
1)Divisibility rule of 2.
If the unit digit of a number is an even number, then the number is divisible by 2.
Ex. – 62682 is divisible by two as its unit digit is an even number.

2)Divisibility rule of 3.
If the sum of the digits of a number is divisible by 3, then the whole number is also divisible by 3.
Ex. – The sum of the digits of the number 261927 is 27, and as 27 is divisible by 3, then 261927 is also divisible by 3.

3)Divisibility rule of 4
If the last two digits of any number are divisible by 4, then the whole number is divisible by 4.
Ex. – The last two digits of the number 272664 are 64, which is divisible by 4. Therefore, the number 272664 is also divisible by 4.

4)Divisibility rule of 5.
If the unit’s digit of a number is either 5 or 0, then the whole number is divisible by 5.
Ex. – The number 2630 is divisible by 5 as its unit’s digit is 0.

5)Divisibility rule of 6.
If a number is divisible by both 2 and 3, then the number is always divisible by 6.
Ex. – 12 is a number which is divisible by both 2 and 3, therefore, it is divisible by 6 also.