Number System Question 2 - CAT 2023

Step-by-Step Explanation

• • Analyze the first factor condition:

  Break down both integers into their respective prime components:

  1134 = 2¹ × 3⁴ × 7¹ ⇒ 1134ⁿ = 2ⁿ × 3⁴ⁿ × 7ⁿ

  168 = 2³ × 3¹ × 7¹

  For 168 to be a factor of 1134ⁿ, every prime exponent of 168 must be less than or equal to the corresponding exponent in 1134ⁿ:

  • Powers of 2: n ≥ 3
  • Powers of 3: 4n ≥ 1 ⇒ n ≥ 1
  • Powers of 7: n ≥ 1

  The least positive integer satisfying these inequalities is n = 3.

  • Analyze the second factor condition:

  Substitute n = 3 back into the base expression:

  1134³ = 2³ × 3¹² × 7³

  168ᵐ = ( 2³ × 3¹ × 7¹ )ᵐ = 2³ᵐ × 3ᵐ × 7ᵐ

  For 1134³ to be a factor of 168ᵐ:

  • Powers of 2: 3m ≥ 3 ⇒ m ≥ 1
  • Powers of 3: m ≥ 12
  • Powers of 7: m ≥ 3

  The least positive integer satisfying these inequalities is m = 12.

  Sum = m + n = 12 + 3 = 15

  Correct Option: B