Question

Starting with 54 and 36, replace the 54 with the greatest common divisor of 54 and 36, obtaining a1, and then replace 36 with the least common multiple of 54 and 36, obtaining b1. Repeat the procedure on a1 and b1, obtaining a2 and b2 and so on. Starting with a1 and b1, what is the least number of times this procedure can be repeated until a1 and b1 are obtained again?

Answer 1

1 time

Explanation:

Given numbers are 54 and 36.

The greatest common divisor of 54 and 36 = 18

So,
`a_1 = 18`.

The least common multiple of 54 and 36 = 108

So,
`b_1 = 108`.

As 54 is replaced by
`a_1` and 36 is replaced by
`b_1`, so after applying the given procedure, the new number is

`(a_1, b_1)=(18, 108)\cdots(i)`

Now, apply the same procedure, to get
`a_2` and
`b_2`.

The greatest common divisor of 18 and 108 = 18

So,
`a_2 = 18`.

The least common multiple of 18 and 108 = 108

So,
`b_2 = 108`.

As
`a_1` is replaced by
`a_2` and
`b_1` is replaced by
`b_2`, so after applying the given procedure, the new number is

`(a_2, b_2)=(18, 108)` which is the same as in equation (i)

Hence, after applying the procedure 1 time after
`(a_1, b_1)`, the obtained number
`(a_2, b_2)` is the same as
`(a_1,b_1)`.