High School

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The H.C.F. of two numbers is 12, and their difference is 12. Which of the following can be the numbers?

A) 70, 84
B) 84, 96
C) 66, 77
D) 80, 92

Answer :

Final answer:

To find the correct pair of numbers with an H.C.F. of 12 and a difference of 12, both numbers must be multiples of 12 with the specified difference. Only option (b) with numbers 84 and 96 meets this criteria.

Explanation:

To determine which pair of numbers has an H.C.F. (Highest Common Factor) of 12 and a difference of 12, we need to apply the fundamentals of number theory. For two numbers to have an H.C.F. of 12, they must both be multiples of 12. Since their difference is also 12, we can express the larger number as 'x' and the smaller one as 'x - 12'. Upon dividing each potential pair of numbers by 12, we should find that both numbers in the valid pair are whole numbers, and their difference should indeed be 12.

  • For option (a) 70 and 84, neither is a multiple of 12 since 70/12 is not a whole number.
  • For option (b) 84 and 96, both are multiples of 12 and their difference is 96 - 84 = 12.
  • For option (c) 66 and 77, neither is a multiple of 12 since 77/12 is not a whole number.
  • For option (d) 80 and 92, neither is a multiple of 12 since both numbers do not give a whole number when divided by 12.

Thus, the correct answer is option (b) 84, 96.

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