If 7-digit number 678p37q is divisible by 75 and p is not a composite, then the values of p and q are:

Given:

7-digit number 678p37q is divisible by 75 and p is not a composite.

Concept:

Divisibility law of 5: A number divisible by 5 if its last digit must be either 0 or 5.

Divisibility law of 3: A number is divisible by 3 if sum of its digit is divisible by 3.

Calculation:

⇒ 7 digit number 678p37q is divisible by 75 means number divisible by 3 and 5 both.

Divisibility rule of 3 apply on 7 digit number 678p37q , then,

⇒ 678p37q should be divisible by 3, then sum of its digit

⇒ 6 + 7 + 8 + p + 3 + 7 + q 

⇒ 31 + p + q

Next values which divisible by 3 i.e 33, 36 ,39. .....(1)

Means p + q can be 2, 5, 8

Now,

Divisibility rule of 5 apply on 7 digit number 678p37q. Then,

q = 0 or 5

If we take q = 0, then p = 2 or 5 or 8. These values of p not given in any option. so,

The value of q must be 5.

Now,

31 + p + 5 

⇒ p + 36 

According to equation (1), next value can be 39. so,

⇒ p + 36 = 39

 =>  p = 3

∴ The value of p and q are 3 and 5 respectively.

Alternate Method

A number is divisible by 75, which means the number is divisible by 25 and 3

If last 2-digit of a number is divisible by 25, then the number is divisible by 25, so q = 5 [ as 75 divisible by 25]

If sum of all the digits divisible by 3, then the number also divisible by 3, so (6 + 7 + 8 + p + 3 + 7 + 5) = 36 + p, here p = 3 [as 6,  9 are composite numbers]

∴ The value of p = 3 and q = 5