Question

think of two numbers whose greatest common factor is 12. If you divide the lesser of the two numbers by that greatest common factor, you get one sixteenth of the other numbers.what are the numbers?

Answer 1

Let two numbers 12n and 12m whose greatest common factor is 12 .[ Here m and n are non divisible by each other and also m > n ]
Bigger number = 12m
and lesser number = 12n

A/C to question,
Lesser number/greatest common factor = 1/16 × other number [ Bigger number]
⇒ 12n/12 = 12m/16
⇒ 16n = 12m
⇒ 4n = 3m
⇒ n /m = 3/4
∵ n and m are non divisible by each other .
∴ n = 3 and m = 4
Now , lesser number = 12n = 12 × 3 = 36
Bigger number = 12m = 12× 4 = 48

Hence, two numbers are 36 and 48

Answer 2

Let the two numbers whose greatest common Factor is 12 be 12a and 12b.

Also, In this,  a and b does not have the any common factor other than on & a > b.
∴ Less number = 12b & and bigger number = 12a.

Now,
From the Question,
Less number/Greatest Common Factor = 1/16 × Bigger Number.
∴ 12b/12 = 1/16 × 12a
⇒  b = 3a/4
⇒ 4b = 3a
⇒ a/b = 4/3

Since, a and b does not have the common factor other than 1.
∴ a = 4 & b = 3.

Greater Number = 12a
= 12 × 4
= 48

Less Number = 24b
= 12 × 3
= 36

Hence, the two numbers are 36 (lesser number) and 48 (bigger number).

[Note ⇒ While solving such Question, the Greatest common factor must be taken as the given number itself, because the greatest factor of that number will be the number itself.]


Hope it helps.