Question

A boy was asked to find the LCM of 3,5,12 and another number.But while calculating, he wrote 21instead of 12and yet came with the correct answer. what could be the fourth number?

Answer 1

Let the fourth number be 'a'
Suppose LCM of 3, 5, 12 and a = x
And LCM of 3, 5, 21, and a = y
Using prime factorization,
12 = 2 × 2 × 3
21 = 3 × 7
Given : x = y
So, a should contain the factors 2 × 2 ×7
Thus, a = 2 × 2 × 7 = 28
Again using prime factorization,
3 = 3
5 = 5
12 = 2 × 2 × 3
21 = 3 × 7
28 = 2 × 2 × 7
So, L. C. M. of 3, 5, 12, and 28 = 3 × 5 × 2 × 2 × 7 = 420
L. C. M. of 3, 5, 21 and 28 = 3 × 5 × 2 × 2 × 7 = 420
So, the fourth number is 28




The prime factors of 3, 5, 12 are 3, 4, 5
and the prime factors for 3, 5, 21 are 3, 5, 7
In order for the LCM to be the same the prime factors have to be the same. but in the first case 7 is missing and in the second case 4 is missing. So, the answer should be a product of these two numbers ie 28.
Now, with 28, the prime factors for 3, 5, 12, 28 are 3, 4, 5, 7 and the LCM will be 3 * 4* 5 * 7 = 420
and as with 21 the prime factors for 3, 5, 21, 28 are 3, 4, 5, 7 and the LCM again is 3 *4 * 5* 7 = 420
Hence 28 is the answer.
Note: There are many answers to the question and 28 is the smallest possible number and all the multiples of 28 are possible solutions for this question.