Number System Question 1 - CAT 2023

Step-by-Step Explanation

• Factorize the base number:

Perform prime factorization on the composite base value:

144 = 12² = ( 2⁴ × 3² )

Raise this structure to the given total exponent:

144¹⁴⁵ = ( 2⁴ × 3² )¹⁴⁵ = 2⁵⁸⁰ × 3²⁹⁰

We are given the baseline structural identity:

aᵐbⁿ = 2⁵⁸⁰ × 3²⁹⁰

• Optimize the variable gap (n - m):

To maximize the expression n - m, maximize the value of n and minimize the value of m.

Since b > 1, assign the largest possible index value to n by matching it directly to the highest prime exponent available, which is 580. Choosing b = 2 yields nₘₐₓ = 580.

This leaves the remaining prime block for aᵐ:

aᵐ = 3²⁹⁰

To minimize the exponent m under the constraint a > 1, condense the entire block into the single largest possible base value: a = 3²⁹⁰, which gives mₘᵢₙ = 1.

Largest Value of n - m = 580 - 1 = 579

Correct Option: D

💡 FundaMakers Tip:

When maximizing the difference between two exponential components (n - m) derived from a prime pool, assign the absolute largest available prime power to the positive component variable (n) and condense the remaining numbers into a single large base to drop the negative component variable (m) down to 1.