Number System Question 6 - CAT 2023
Question
The number of coins collected per week by two coin-collectors A and B are in the ratio 3 : 4. If the total number of coins collected by A in 5 weeks is a multiple of 7, and the total number of coins collected by B in 3 weeks is a multiple of 24, then the minimum possible number of coins collected by A in one week is:
Step-by-Step Explanation
- Set up the weekly ratio tracking variables:
Let A collect 3c coins per week and B collect 4c coins per week, where c is a positive integer scalar.
- Apply the weekly multiple constraints:
- For Collector A (5 weeks): Total coins = 5 × 3c = 15c. This value must be a multiple of 7.
Since gcd(15,7) = 1, the variable c must be a multiple of 7.
- For Collector B (3 weeks): Total coins = 3 × 4c = 12c. This value must be a multiple of 24.
12c = 24k ⇒ c = 2k ⇒ c must be a multiple of 2.
- Determine the minimum valid value:
To minimize the number of coins, find the least common multiple of the two independent constraints on c:
cₘᵢₙ = LCM(7,2) = 14
Minimum weekly coins collected by A = 3cₘᵢₙ = 3 × 14 = 42
Correct Answer: 42