Number System Question 2 - CAT 2022

Step-by-Step Explanation

• Find the value of A:

We need to find the largest number that divides 3ᵏ + 4ᵏ + 5ᵏ no matter what positive number we choose for k. Let's test this using the simplest values for k:

• If k = 1, the expression becomes 3¹ + 4¹ + 5¹ = 12.
• If k = 2, the expression becomes 3² + 4² + 5² = 9 + 16 + 25 = 50.

The largest number that can divide both 12 and 50 is 2. So, A = 2.

• Find the value of B:

The second expression is 4ᵏ + 3( 4ᵏ ) + 4ᵏ⁺². Let's simplify this using basic counting. Think of 4ᵏ as a single box.

Expression = 1 box + 3 boxes + 4ᵏ⁺²

Since 4ᵏ⁺² = 4ᵏ × 4² = 16 boxes, we can add them all up:

Total = 1 + 3 + 16 = 20 boxes = 20 × 4ᵏ

Now let's test this simplified expression with simple values of k:

• If k = 1, it becomes 20 × 4¹ = 80.
• If k = 2, it becomes 20 × 4² = 320.

The largest number that can cleanly divide both 80 and 320 is 80. So, B = 80.

Final Answer = A + B = 2 + 80 = 82

Correct Answer: 82

💡 FundaMakers Tip:

Don't let letters in the exponents scare you! When a question says a rule works "for any positive integer k", it means you can pick the easiest numbers on earth. Just plug in k = 1 and k = 2, find the highest common factor (HCF) of those results, and you will have your answer in no time.