To find the HCF of 65 and 117,

117 = 1×65 + 52

65 = 1× 52 + 13

52 = 4 ×13 + 0

therefore HCF of 65 and 117 is 13.

65m - 117 = 13

65m = 117+13 = 130

∴m =130/65 = 2

Now, 65 = 5 × 13

117 = 13 × 3 × 3

LCM of 65 and 117 = 5 × 13 × 3 × 3 = 585.

Find m in HCF expression

To find the value of m in the expression 65m - 117, we need to use the Euclidean algorithm to find the HCF of 65 and 117.

Euclidean Algorithm

Step 1: Divide the larger number by the smaller number and find the remainder.
117 divided by 65 gives a remainder of 52.

Step 2: Divide the smaller number (65) by the remainder (52) and find the remainder.
65 divided by 52 gives a remainder of 13.

Step 3: Divide the previous remainder (52) by the new remainder (13) and find the remainder.
52 divided by 13 gives a remainder of 0.

The HCF of 65 and 117 is the last non-zero remainder, which is 13.

Substituting the value of HCF (13) in the expression 65m - 117, we get:
13 = 65m - 117
Solving for m, we get:
m = 8

Therefore, the value of m in the expression 65m - 117 is 8.

Find LCM using prime factor method

To find the LCM of 65 and 117 using the prime factor method, we need to find the prime factors of both numbers.

Step 1: Prime factors of 65
65 = 5 x 13

Step 2: Prime factors of 117
117 = 3 x 3 x 13

Step 3: Write down all the prime factors of both numbers.
Prime factors of 65: 5, 13
Prime factors of 117: 3, 3, 13

Step 4: Identify the common prime factors and write them down once.
Common prime factors: 3, 13

Step 5: Multiply the common prime factors and the remaining prime factors.
LCM = 3 x 3 x 5 x 13
LCM = 585

Therefore, the LCM of 65 and 117 is 585.