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15 Feb 2026, 06:08
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C
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
55%
(hard)
Question Stats:
65% (02:05) correct
35%
(02:07)
wrong
based on 66
sessions
History
Date
Time
Result
Not Attempted Yet
A marketing class with 242 male students and 198 female students is to be divided into n groups, each containing m male students and f female students. What is the smallest possible value of m + f ?
A. 9
B. 11
C. 20
D. 30
E. 31
A. 9
B. 11
C. 20
D. 30
E. 31
15 Feb 2026, 09:06
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Key Concept Being Tested: Greatest Common Divisor (GCD) and divisibility constraints
Step-by-step Solution:
1. Understand the constraint: Each group must have the same composition: m males and f females. This means we need to divide 242 males into groups of m, and 198 females into groups of f, with the same number of groups (n) for both.
2. Set up the equations:
Number of groups = 242/m = 198/f = n
Both divisions must result in whole numbers (can't split students!)
3. Find what values work: For this to work, m must be a divisor of 242, and f must be a divisor of 198. Since we need the same number of groups: 242/m = 198/f
4. Rewrite the relationship: This means 242f = 198m, which simplifies to 121f = 99m. Reducing further: 11f = 9m
5. Find the smallest values: The smallest positive integers satisfying 11f = 9m are f = 9 and m = 11.
Check: 11(9) = 99 ✓ and 9(11) = 99 ✓
6. Verify the solution:
Number of groups with m = 11: 242 ÷ 11 = 22 groups
Number of groups with f = 9: 198 ÷ 9 = 22 groups ✓ (matches!)
Each group has: m + f = 11 + 9 = 20 students
7. Check smaller options: Could m + f = 9? No, because we'd need m and f values that satisfy our equation where both sum to 9, but f = 9, m = 0 doesn't work (we need male students). Similarly, m + f = 11 doesn't yield valid integer solutions that divide evenly.
Answer: C (20)
Common Trap: Students often try to simply find GCD(242, 198) = 22, thinking that's the group size. But the question asks for m + f (students per group), not the number of groups. The GCD actually tells us how many groups we can form!
Takeaway: When dividing items into equal groups with multiple constraints, look for the relationship between divisors that satisfies all conditions simultaneously—use the ratio between total quantities to find the minimum valid values.
Step-by-step Solution:
1. Understand the constraint: Each group must have the same composition: m males and f females. This means we need to divide 242 males into groups of m, and 198 females into groups of f, with the same number of groups (n) for both.
2. Set up the equations:
Number of groups = 242/m = 198/f = n
Both divisions must result in whole numbers (can't split students!)
3. Find what values work: For this to work, m must be a divisor of 242, and f must be a divisor of 198. Since we need the same number of groups: 242/m = 198/f
4. Rewrite the relationship: This means 242f = 198m, which simplifies to 121f = 99m. Reducing further: 11f = 9m
5. Find the smallest values: The smallest positive integers satisfying 11f = 9m are f = 9 and m = 11.
Check: 11(9) = 99 ✓ and 9(11) = 99 ✓
6. Verify the solution:
Number of groups with m = 11: 242 ÷ 11 = 22 groups
Number of groups with f = 9: 198 ÷ 9 = 22 groups ✓ (matches!)
Each group has: m + f = 11 + 9 = 20 students
7. Check smaller options: Could m + f = 9? No, because we'd need m and f values that satisfy our equation where both sum to 9, but f = 9, m = 0 doesn't work (we need male students). Similarly, m + f = 11 doesn't yield valid integer solutions that divide evenly.
Answer: C (20)
Common Trap: Students often try to simply find GCD(242, 198) = 22, thinking that's the group size. But the question asks for m + f (students per group), not the number of groups. The GCD actually tells us how many groups we can form!
Takeaway: When dividing items into equal groups with multiple constraints, look for the relationship between divisors that satisfies all conditions simultaneously—use the ratio between total quantities to find the minimum valid values.
poojaarora1818
Joined: 30 Jul 2019
Last visit: 24 Apr 2026
Posts: 1,624
Given Kudos: 3,818
Location: India
Concentration: General Management, Economics
Schools: Duke '27 Haas '27 Kelley '27 Kellogg '27 McDonough '27
GPA: 3
WE:Human Resources (Real Estate)
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15 Feb 2026, 11:12
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Solution:
I tried to solve it by finding the GCD of 242(males) and 198(females).
242 = 2 * 11^2
198 = 2 * 11 * 3^2
GCD of 242 and 198 is 2 * 11 = 22
As we know, 242/n + 198/n = 440/n
Males per group = 242/22 = 11
Females per group= 198/22= 9
Males + females = 11 + 9 = 20
Answer: C
I tried to solve it by finding the GCD of 242(males) and 198(females).
242 = 2 * 11^2
198 = 2 * 11 * 3^2
GCD of 242 and 198 is 2 * 11 = 22
As we know, 242/n + 198/n = 440/n
Males per group = 242/22 = 11
Females per group= 198/22= 9
Males + females = 11 + 9 = 20
Answer: C
kevincan
