Final answer:
The possible numbers whose LCM is 130 and HCF is 13, and which are less than 100, are 26 and 65.
Step-by-step explanation:
The question is asking to list the possible numbers whose Least Common Multiple (LCM) is 130 and whose Highest Common Factor (HCF) is 13, with both numbers being less than 100. Knowing that the LCM of two numbers is equal to the product of these numbers divided by their HCF (LCM = a * b / HCF), we can use this information to solve the problem.
Let the two numbers be a and b. Since LCM(a, b) = 130 and HCF(a, b) = 13, we can write:
a * b = LCM * HCF
a * b = 130 * 13
a * b = 1690
Now we need to find the pairs of factors of 1690 that are less than 100. The factor pairs of 1690 are (1, 1690), (2, 845), (5, 338), (10, 169), (13, 130), (26, 65).
Since both numbers must be less than 100 and share a HCF of 13, we discard the pairs where one or both factors exceed 100, and those that don't share 13 as a factor. This leaves us with the pair (26, 65), as both numbers are less than 100 and share 13 as the HCF.
Therefore, the possible numbers are 26 and 65.