GMAT Focus EditionQuantitative ReasoningHard
A certain positive integer 'x' leaves a remainder of 2 when divided by 3, and a remainder of 3 when divided by 4. What is the smallest possible value of 'x'?
- A14
- B7
- C23
- D11
Show answer & explanationAnswer & explanation
Correct answer: D. 11
From the first condition, x = 3k + 2. From the second condition, x = 4m + 3. We are looking for the smallest 'x'. If k=0, x=2 (2 mod 4 = 2, not 3) If k=1, x=5 (5 mod 4 = 1, not 3) If k=2, x=8 (8 mod 4 = 0, not 3) If k=3, x=11 (11 mod 4 = 3). This satisfies both conditions. So the smallest value of x is 11.
Why the other options are wrong
- A. 14 leaves a remainder of 2 when divided by 3, but a remainder of 2 when divided by 4 (not 3).
- B. 7 leaves a remainder of 1 when divided by 3 (not 2).
- C. 23 leaves a remainder of 2 when divided by 3, and a remainder of 3 when divided by 4, but it is not the smallest possible value (11 is smaller).
Chinese Remainder Theorem (Basic Application)
A theorem that provides a unique solution (modulo the product of the moduli) to a system of simultaneous congruences.
- Involves finding a number that leaves specific remainders when divided by different numbers.
- Often solved by listing possibilities or using number theory principles.
- The smallest positive integer is usually sought.
Memory trick: List, check, intersect: find the number that fits all remainder rules.