Two integers have H.C.F. (greatest common divisor) equal to 12, and the difference between the two numbers is also 12. Which pair of numbers satisfies both conditions?

Introduction / Context:
We are given two constraints for a pair of integers: their highest common factor (HCF) is 12, and their difference is 12. We must select the pair that meets both conditions at once.

Given Data / Assumptions:

• HCF = 12.
• Absolute difference between the numbers = 12.
• All choices are positive integer pairs.

Concept / Approach:
If HCF is 12, then each number is a multiple of 12. Let the numbers be 12m and 12n, with gcd(m, n) = 1 (since all common factors are captured by 12). The difference is |12m − 12n| = 12|m − n| and is given as 12 ⇒ |m − n| = 1, so m and n must be consecutive coprime integers.

Step-by-Step Solution:

Verification / Alternative check:
Factor 84 = 2^2*3*7 and 96 = 2^5*3; common prime factors have minimum exponents 2^2*3 = 12. Difference is 12. Both constraints satisfied.

Why Other Options Are Wrong:

• 66, 78; 70, 82; 94, 106: Correct difference but HCF is not 12.
• 72, 84 (added distractor): HCF is 12, but difference is also 12; however gcd(72,84)=12 is true; if present as an option it would also satisfy. Among the provided original options, 84, 96 is the correct listed pair.

Common Pitfalls:

• Forgetting that both numbers must be multiples of 12.
• Not checking gcd precisely, especially when both are even.

Final Answer:
84, 96