Multiple choice

How many distinct pairs of 2-digit numbers are possible which have their HCF as 15?

  1. 15

  2. 18

  3. 13

  4. 11

  5. 12
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Let the numbers be 15a and 15b where a and b are coprime. For 2-digit numbers, 15a < 100 and 15b < 100, so a, b are in {1, 2, 3, 4, 5, 6}. Pairs (a,b) with gcd(a,b)=1 are (1,2), (1,3), (1,4), (1,5), (1,6), (2,3), (2,5), (3,4), (3,5), (4,5), (5,6). There are 11 such pairs.

AI explanation

Since the highest common factor of the two numbers is 15, let the two distinct numbers be 15p and 15q, where p and q are co-prime natural numbers. Because the numbers are two-digit, they must be less than 100, meaning 15p < 100 and 15q < 100, which implies p and q can only be from the set of numbers from 1 to 6. We must find co-prime pairs from this set: (1,2), (1,3), (1,4), (1,5), (1,6), (2,3), (2,5), (3,4), (3,5), (4,5), and (5,6). Counting these 11 distinct pairs gives the result 11.

Keep practicing — related questions