L.C.M of two numbers is 234 whereas H.C. F of the same pair is 13. Find…

To find the smallest factor of the numbers given their Least Common Multiple (LCM) and Highest Common Factor (HCF), we use the fundamental property that the product of two numbers is equal to the product of their LCM and HCF.

Step-by-Step Calculation
Understand the Relationship:
The product of two numbers is equal to HCF * LCM.
Product = 13 * 234 = 3042.

Analyze the Parity (Even/Odd):
The product 3042 is an even number. For the product of two numbers to be even, at least one of the numbers must be even. An even number is, by definition, divisible by 2.

Identify the HCF Constraint:
Both numbers must be multiples of the HCF, which is 13.
If we let the two numbers be 13a and 13b (where a and b are coprime), their product is:
(13 * a) * (13 * b) = 3042
169 * (a * b) = 3042
a * b = 3042 / 169 = 18

Now we find coprime pairs (a, b) whose product is 18:

Pair (1, 18): Numbers are (13 * 1) = 13 and (13 * 18) = 234.

Pair (2, 9): Numbers are (13 * 2) = 26 and (13 * 9) = 117.

Determine the Smallest Factor:

In the first pair (13, 234), the factors are 13, 2, 3, etc.

In the second pair (26, 117), the factors are 2, 13, 3, etc.

Comparing these, 2 is the smallest prime factor present in both sets of numbers.