Question

the HCF and lcm of two numbers are 6 and 336 respectively. find the two numbers, if the difference between them is 6​

Answer 1

Let’s break this down a bit.

HCF(a,b)=6 . That means that:

6 divides a and 6 divides b and that 6 is the highest number for which this is true.

LCM(a,b)=336 . That means that:

a divides 336 and b divides 336 and 336 is the lowest number for which this is true.

Now, what does it mean that x divides y ? It means that the set of prime factors of x is a subset of the set of prime factors of y . So let’s get the prime factors of all of the numbers involved here:

336=2×2×2×2×3×7

6=2×3

Now, we have two numbers a and b that are not the same, but must together use all of the prime factors of 336 and no more, and must both contain the prime factors of 6 , but have no more factors in common.

So, let’s start off with a=2×3×… . Now we have two options: Either we can tack on more 2 s to the end, or we can tack on a 7 . Note that if we tack on one 2 , we have to tack on all the 2 s because otherwise b would have to take those 2 s and that would make the common factors larger, and the HCF higher than 6 .

So, let’s set a=2×2×2×2×3 and b=2×3×7 . If you look, they only have 2×3 in common, so HCF(a,b)=2×3=6 and together they cover 2×2×2×2×3×7 , so LCM(a,b)=2×2×2×2×3×7=336 .

So what are these numbers? a=2×2×2×2×3=48 and b=2×3×7=42 .

Now we check, is the difference between the two 6 ? 48−42=6 . Yes.

So, the numbers are 42 and 48 .

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