Question

A and B are the largest two-digit co-prime numbers, whose digits are also co-prime to each other. If A + B is divided by 11, the remainder is r ?

A. r² is a perfect cube

B. r + 1 is a perfect square

C. r is a perfect cube

D. r – 1 is a prime number

1.

Only A is correct

2.

B and D are correct

3.

A, B, and C are correct

4.

All are correct

Answer 1

Correct Answer - Option 4 :

All are correct

Formula Used:

Remainder Theorem

Dividend = (Divisor × Quotient) + Remainder           

Concept Used:

Any set of numbers which do not have any common factor other than 1 are called co-prime or relatively prime numbers.

Calculation:

Largest two-digit co-prime numbers = 99, 98

but digits of 99 are not co-prime to each other.

We consider 98 and 97 both are co-prime and the digits of each number are co-prime to each other.

A + B = 98 + 97 = 195

Using the remainder theorem,

195 = 11 × 17 + 8

⇒ r = 8

r² = 8²

⇒ 64

⇒ 4³

r² is a perfect cube

Option A is correct.

r + 1 = 9

⇒ 3²

r + 1 is a perfect square

Option B is correct.

r = 8

⇒ 2³

r is a perfect cube

Option C is correct.

r – 1 = 8 – 1

⇒ 7

r – 1 is a prime number

Option D is correct.

∴ All are correct.