Which of the following is the greatest number that divides 104 and 133 and leaves 8 and 5 as respective remainders?

Correct option is B

Given:

When 104 is divided by the required number, the remainder is 8.

When 133 is divided by the required number, the remainder is 5.

Formula Used:

The remainder when 104 is divided by the required number, d, is 8.

Hence, 104 = dq₁ + 8, where q₁ is the quotient.

The remainder when 133 is divided by the required number, d, is 5.

Hence, 133 = dq₂ + 5, where q₂ is the quotient.

Subtract the two equations:

(104 - 8) - (133 - 5) = d(q₁ - q₂)

Simplifying: -32 = d(q₁ - q₂)

Solution:

Now, we have to find the greatest divisor d that divides -32.

The divisors of 32 are: 1, 2, 4, 8, 16, 32.

We now check each divisor to see which satisfies the conditions.

For d = 32:

104 ÷ 32 = 3 remainder 8 (satisfies the first condition).

133 ÷ 32 = 4 remainder 5 (satisfies the second condition).

Hence, the greatest number that divides both 104 and 133 and leaves 8 and 5 as respective remainders is 32.