Answer :
Final answer:
The student's question involves finding the larger of two numbers with a given H.C.F of 84, L.C.M of 21, and a ratio of 1:4 between the two numbers. Using the fundamental theorem of arithmetic and setting up an equation with the given ratio, we can solve for the larger number.
Explanation:
The question asks to find the larger of two numbers given their highest common factor (H.C.F) and lowest common multiple (L.C.M), as well as the ratio between them. According to the fundamental theorem of arithmetic, for any two numbers, A and B, the product of the H.C.F and L.C.M is equal to the product of A and B (H.C.F × L.C.M = A × B).
In this case, H.C.F is 84 and L.C.M is 21. Therefore, the product of the two numbers (A × B) is 84 × 21.
The ratio given is 1:4. If we let the smaller number be A and the larger number be B, we can say A:B is 1:4. Hence, A = x, B = 4x.
Now, we have the equation: A × B = 84 × 21 => x × 4x = 84 × 21. By solving this equation, we find x, and subsequently 4x, which is the larger number we are seeking.
