Consider all prime numbers between 1 and 100.

Which of the following statements is (are) correct?

a. A number that is one greater than a multiple of 5 has a unit digit 3 or 6.

b. The sum of all numbers which are one greater than a multiple of 5 is 215.

c. The sum of all numbers which are one greater than a multiple of 5 and also one greater than a multiple of 6 is 92.

Correct option is D

Given:

All prime numbers between 1 and 100.

Prime numbers = 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97
Analyze each statement
Statement a (I): “A number that is one greater than a multiple of 5 has a unit digit 3 or 6.”
​$$5 \times 1 = 5 \Rightarrow 5 + 1 = 6$$​​
​$$5 \times 2 = 10 \Rightarrow 10 + 1 = 11$$​​
​$$5 \times 3 = 15 \Rightarrow 15 + 1 = 16$$​​
​$$5 \times 4 = 20 \Rightarrow 20 + 1 = 21$$​​
​$$5 \times 5 = 25 \Rightarrow 25 + 1 = 26$$​​
Result: The unit digits of numbers one more than a multiple of 5 are 1 or 6, not 3 or 6.
Hence, Statement a is FALSE.
Statement b (II): “The sum of all numbers which are one greater than a multiple of 5 is 215.”
​$$5 \times 2 + 1 = 11$$​​
​$$5 \times 6 + 1 = 31$$​​
​$$5 \times 8 + 1 = 41$$​​
​$$5 \times 12 + 1 = 61$$​​
​$$5 \times 14 + 1 = 71$$​​
These are all primes. So selected numbers: 11, 31, 41, 61, 71
Their sum: $$11 + 31 + 41 + 61 + 71 = 215$$​​
Statement b is TRUE
 Statement c (III):“The sum of all numbers which are one greater than a multiple of 5 and also one greater than a multiple of 6 is 92.”
LCM (5,6) = 30
Form of such numbers: 30k + 1
​$$30 \times 1 + 1 = 31$$​​
​$$30 \times 2 + 1 = 61$$​​
​$$30 \times 3 + 1 = 91$$​​
From these, which are prime?
31 → Prime
61 → Prime
91 → Not Prime
So valid primes: 31 and 61
Their sum: 31 + 61 = 92
Statement c is TRUE

Thus, the correct option is (d) b and c