Among 65 and 117; 117 > 65

Since 117 > 65, we apply the division lemma to 117 and 65 to obtain

117 = 65 × 1 + 52** … Step 1**

Since remainder 52 ≠ 0, we apply the division lemma to 65 and 52 to obtain

65 = 52 × 1 + 13** … Step 2**

Since remainder 13 ≠ 0, we apply the division lemma to 52 and 13 to obtain

52 = 4 × 13 + 0** … Step 3**

In this step the remainder is zero. Thus, the divisor i.e. 13 in this step is the H.C.F. of the given numbers

The H.C.F. of 65 and 117 is 13

From **Step 2**:

13 = 65 – 52 × 1** … Step 4**

From **Step 1**:

52 = 117 – 65 × 1

Thus, from **Step 4**, it is obtained

13 = 65 – (117 – 65 × 1) × 1

⇒13 = 65 × 2 – 117

⇒13 = 65 × 2 + 117 × (–1)

In the above relationship the H.C.F. of 65 and 117 is of the form 65*m* + 117 *n*, where *m* = 2 and *n* = –1

**65=5*13**

**117=3*3*13**

**LCM(65,117)=5*3*3*13**

** =585**