IPMAT Indore 2022 · Quantitative Ability · Question 33

Step 1 — translate the words into algebra. Let the two natural numbers be a and b. The description says:

\frac{{\left( a+b \right)}^{2}-{\left( a-b \right)}^{2}}{4}

Step 2 — simplify. Expanding,

{\left( a+b \right)}^{2}-{\left( a-b \right)}^{2}=\left({a}^{2}+2ab+{b}^{2}\right)-\left({a}^{2}-2ab+{b}^{2}\right)=4ab

so the whole expression equals

\frac{4ab}{4}=ab

Step 3 — express ab in the language of the options. The fundamental identity relating two numbers to their HCF and LCM:

\text{HCF}\left( a,b \right)\times \text{LCM}\left( a,b \right)=ab

So the result is the product of the LCM and HCF.

Check with a=6, b=4: \frac{100-4}{4}=24, and \text{HCF}=2, \text{LCM}=12, product =24 ✔

Answer: Option 1 — the product of the LCM and HCF of the two numbers

Shortcut. The identity {\left( a+b \right)}^{2}-{\left( a-b \right)}^{2}=4ab is the whole question — divide by 4 and you have ab.

Then recall \text{HCF}\times \text{LCM}=ab, which holds for any two positive integers.

Test the options with one example if you want certainty fast: a=6, b=4 gives 24. The HCF alone is 2, the LCM alone is 12, and {\left( ab \right)}^{2}=576 — so only option (a) matches. One numerical example separates all four options here, which is usually the quickest route when the options are verbal descriptions.