Question

Suppose a and b are positive integers such that lcm(a,b) = 90 and gcd(a,b) = 3. If b is three more than a, then what are the values of a and b?​

Answer 1

Given:

LCM(a,b) = 90 and GCD(a,b) = 3.

b is three more than a.

To find:

The values of a and b.

Solution:

We have,

LCM(a,b) = 90

GCD(a,b) = HCF(a,b) = 3

According to the question,

`b=a+3`

If a and b are two positive integers, then

`a* b=HCF(a,b)* LCM(a,b)`

`a* (a+3)=3* 90`

`a^2+3a=270`

`a^2+3a-270=0`

Splitting the middle terms, we get

`a^2+18a-15a-270=0`

`a(a+18)-15(a+18)=0`

`(a+18)(a-15)=0`

Using zero product property, we get

`a+18=0` and
`a-15=0`

`a=-18` and
`a=15`

a is a positive integer so it cannot be negative. So, a=15.

Now,

`b=a+3`

`b=15+3`

`b=18`

Therefore, the value of a is 15 and the value of b is 18.