Number System Question 9 - CAT 2020
If a, b and c are positive integers such that ab= 432 , bc = 96 and c < 9, then the smallest possible value of a + b + c is:
Step-by-Step Explanation
We are given the following conditions for the positive integers a,b, and c:
- ab = 432
- bc = 96
- c < 9
Notice that b is a common factor in both product terms. To minimize the sum a + b + c, we need to select a value for the shared middle variable b that keeps all three numbers balanced and close to each other.
Let's test the factors of 96 where c < 9 and see which ones yield integer values for a:
- If c = 2 ⇒ b = (96)/(2) = 48 ⇒ a = (432)/(48) = 9.
Sum = 9 + 48 + 2 = 59
- If c = 4 ⇒ b = (96)/(4) = 24 ⇒ a = (432)/(24) = 18.
Sum = 18 + 24 + 4 = 46
- If c = 6 ⇒ b = (96)/(6) = 16 ⇒ a = (432)/(16) = 27.
Sum = 27 + 16 + 6 = 49
- If c = 8 ⇒ b = (96)/(8) = 12 ⇒ a = (432)/(12) = 36.
Sum = 36 + 12 + 8 = 56
Comparing the sums, the smallest value is 46.
Correct Option: D
đź’ˇ FundaMakers Tip:
To minimize the sum of multiplied terms, look for factor combinations that bring the numbers as close to each other as possible. Testing the valid values for c (2,4,6,8) reveals the minimum point quickly without long calculations.