The L.C.M. of two numbers is 10 times their H.C.F. Find the numbers.

To solve this problem, let's first understand what L.C.M. and H.C.F. mean.

L.C.M. (Least Common Multiple): The L.C.M. of two or more numbers is the smallest number that is divisible by all the given numbers.

H.C.F. (Highest Common Factor): The H.C.F. of two or more numbers is the largest number that divides each of the given numbers without leaving a remainder.

Given that the L.C.M. of two numbers is 10 times their H.C.F., we can express this mathematically as:

L.C.M. = 10 * H.C.F.

Now, let's solve the problem using the given answer choices:

a) 24, 96:
The H.C.F. of 24 and 96 is 24.
L.C.M. = 10 * H.C.F. = 10 * 24 = 240
However, the L.C.M. of 24 and 96 is not equal to 240, so this option is incorrect.

b) 16, 15:
The H.C.F. of 16 and 15 is 1.
L.C.M. = 10 * H.C.F. = 10 * 1 = 10
However, the L.C.M. of 16 and 15 is not equal to 10, so this option is incorrect.

c) 24, 480:
The H.C.F. of 24 and 480 is 24.
L.C.M. = 10 * H.C.F. = 10 * 24 = 240
The L.C.M. of 24 and 480 is equal to 240, so this option is a possible answer.

d) 24, 240:
The H.C.F. of 24 and 240 is 24.
L.C.M. = 10 * H.C.F. = 10 * 24 = 240
The L.C.M. of 24 and 240 is equal to 240, so this option is a possible answer.

Since both options c) and d) give the L.C.M. as 240, we need to check the H.C.F. to determine the correct answer.

The H.C.F. of 24 and 480 is 24, while the H.C.F. of 24 and 240 is also 24.

Therefore, the correct answer is option d) 24, 240, as it satisfies both the condition of having the L.C.M. as 240 and the H.C.F. as 24.

Summary:
The L.C.M. of two numbers is 10 times their H.C.F. The correct answer is option d) 24, 240, as it satisfies both the condition of having the L.C.M. as 240 and the H.C.F. as 24.