Using HCF and LCM to identify the larger number

Key takeaway: This is a number theory problem where the sum alone is not enough, so we combine the HCF and LCM information. The standard strategy is to write the numbers as multiples of their HCF and use the product relation for two numbers with a known HCF and LCM.

Set up the two numbers

Let the numbers be $27m$ and $27n$, since their HCF is 27. Because the HCF is 27, $m$ and $n$ must be coprime.

We are also told the sum is 135:

$27m+27n=135$

Factor out 27:

$27(m+n)=135$

So

$m+n=5.$

Use the LCM information

For two numbers of the form $27m$ and $27n$ with $m$ and $n$ coprime, the LCM is

$27mn.$

We are told the LCM is 162, so

$27mn=162$

which gives

$mn=6.$

Now we need two coprime positive integers with

$m+n=5 \quad \text{and} \quad mn=6.$

The pair is $m=2$ and $n=3$.

So the two numbers are

$27\cdot 2=54 \quad \text{and} \quad 27\cdot 3=81.$

Final answer

The larger number is

$\boxed{81}.$

Why this works

The HCF lets us factor out the common part of both numbers, and the LCM helps confirm the coprime multipliers. Once the numbers are reduced to a simple pair of integers, the problem becomes a small arithmetic search instead of a trial-and-error guess.

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Pitfalls the pros know 👇 A common mistake is to use the formula $\text{product} = \text{HCF} \times \text{LCM}$ without checking whether the given sum is consistent with the pair. Here the sum and LCM must both match the same pair of numbers, so you should reduce the problem to coprime factors after dividing by the HCF. Another error is to assume the two numbers are 27 and 108 just because they add to 135; that pair has HCF 27 but the LCM is not 162. Always verify both conditions together, not one at a time.

What if the problem changes? If the sum were 189 instead of 135 while the HCF remained 27, you would still write the numbers as $27m$ and $27n$. Then $m+n=7$, and if the LCM were changed to 486, you would get $27mn=486$, so $mn=18$. The solution would then come from the coprime factor pair matching both conditions. This shows that every version of the problem follows the same pattern: divide by the HCF, solve the small integer system, then scale back up.

Tags: greatest common divisor, least common multiple, coprime integers