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If the HCF of 65 and 117 is Expressible in the form 65m – 117, Calculate the Value of m

 Q. If the HCF of 65 and 117 is expressible in the form 65m – 117, calculate the value of m?

Solution: The highest common factor is defined as the largest common factor that two or more numbers have. Let us first calculate the HCF of 65 and 117 following the below listed steps:

Step 1: Find all the prime factors of the given numbers i.e., 65 and 117

Number

Pairs of prime factors

Factors

65

1 x 65

5 x 13

1, 5, 13, 65

117

1 x 117

3 x 3 x 13

1, 3, 13, 117

Step 2: Identifying the common factors among the prime factors of the given numbers, we find that the Common Factors of 65 and 117 are 1 and 13.

Step 3: The largest of the common factors of the two numbers is the highest common factor (HCF).

Here, the largest of the common factors of 65 and 117 is 13.  Thus, the highest common factor (HCF) of 65 and 117 is 13.

Now, it is given that the HCF of 65 and 117 is expressible in form 65m – 117. So, let us equate it with the HCF to find the value of m.

Therefore, 13 = 65m – 117 

               Or 13 + 117 = 65m

               Or 130 = 65m

               Or 130/65 = m

                Or m = 2 

Hence the value of m as per the given equation is 2.

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