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Harrizon [31]
2 years ago
11

Starting with 54 and 36, replace the 54 with the greatest common divisor of 54 and 36, obtaining a1, and then replace 36 with th

e 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?
Mathematics
1 answer:
nalin [4]2 years ago
6 0

Answer:

1 time

Step-by-step 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).

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