To find the highest common factor (HCF) of 65 and 117, we can use the method of prime factorization or the Euclidean algorithm. Here, we’ll use the Euclidean algorithm for simplicity.
Step 1: Apply the Euclidean algorithm.
We divide the larger number (117) by the smaller number (65) and find the remainder:
117 ÷ 65 = 1 remainder 52
Step 2: Now, we take the divisor (65) and the remainder (52) and repeat the process:
65 ÷ 52 = 1 remainder 13
Step 3: Repeat again with 52 and 13:
52 ÷ 13 = 4 remainder 0
Since we reached a remainder of 0, the last non-zero remainder is 13. Therefore, the HCF of 65 and 117 is 13.
Now, we need to find integral values of m and n such that HCF(65m, 117n) = 13.
To produce 13 as the highest common factor of 65m and 117n, we can set:
- m = 1 and n = 1. Thus, HCF(65 * 1, 117 * 1) = HCF(65, 117) = 13.
Alternatively, we can choose any pair where m and n multiply with their respective coefficients in such a way that they retain 13 as a common factor. For example:
- m = 2 and n = 1 (HCF(130, 117) = 13)
- m = 1 and n = 2 (HCF(65, 234) = 13)
Therefore, some integral pairs (m, n) could be (1, 1), (2, 1), or (1, 2). In this way, we can establish different combinations that yield an HCF of 13.