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06 Feb 2026, 16:11
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Difficulty:
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Question Stats:
23% (02:11) correct
78%
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wrong
based on 40
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For two distinct natural numbers x and y, the sum of the reciprocals of these numbers is 1/3. What is the minimum possible value of x + y?
A. 8
B. 12
C. 14
D. 16
E. 18
(Source=TCYOnline)
A. 8
B. 12
C. 14
D. 16
E. 18
(Source=TCYOnline)
06 Feb 2026, 23:25
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Great question — this one tests number theory + fraction manipulation, and the trap is trying random pairs instead of setting up the algebra cleanly.
Step 1: Set up the equation.
We're told 1/x + 1/y = 1/3, where x and y are distinct natural numbers.
Rewriting: (x + y) / (xy) = 1/3, which gives us 3(x + y) = xy.
Step 2: Rearrange to isolate one variable.
3x + 3y = xy
xy − 3x − 3y = 0
Add 9 to both sides (Simon's Favorite Factoring Trick):
xy − 3x − 3y + 9 = 9
(x − 3)(y − 3) = 9
Step 3: Find factor pairs of 9.
Since x and y are distinct natural numbers, (x − 3) and (y − 3) must be positive integer factor pairs of 9:
- (1, 9) → x = 4, y = 12 → x + y = 16
- (3, 3) → x = 6, y = 6 → rejected (x and y must be distinct)
- (9, 1) → x = 12, y = 4 → x + y = 16 (same pair, just swapped)
Step 4: Verify.
1/4 + 1/12 = 3/12 + 1/12 = 4/12 = 1/3 ✓
The minimum value of x + y = 16.
Common trap: Many students try to guess values like (4, 12) or (6, 6) without a system — this wastes time and risks missing cases. The real trap here is the pair (6, 6), which satisfies the reciprocal equation but violates the "distinct" constraint. Always re-read the stem.
Takeaway: Whenever you see 1/x + 1/y = 1/k with integer constraints, reach for Simon's Favorite Factoring Trick — rearrange, add a constant to both sides, and factor. It turns an open-ended search into a clean factor-pair problem every time.
Answer: D (16)
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Free gamified GMAT prep: edskore.com | 725 GMAT Focus
Step 1: Set up the equation.
We're told 1/x + 1/y = 1/3, where x and y are distinct natural numbers.
Rewriting: (x + y) / (xy) = 1/3, which gives us 3(x + y) = xy.
Step 2: Rearrange to isolate one variable.
3x + 3y = xy
xy − 3x − 3y = 0
Add 9 to both sides (Simon's Favorite Factoring Trick):
xy − 3x − 3y + 9 = 9
(x − 3)(y − 3) = 9
Step 3: Find factor pairs of 9.
Since x and y are distinct natural numbers, (x − 3) and (y − 3) must be positive integer factor pairs of 9:
- (1, 9) → x = 4, y = 12 → x + y = 16
- (3, 3) → x = 6, y = 6 → rejected (x and y must be distinct)
- (9, 1) → x = 12, y = 4 → x + y = 16 (same pair, just swapped)
Step 4: Verify.
1/4 + 1/12 = 3/12 + 1/12 = 4/12 = 1/3 ✓
The minimum value of x + y = 16.
Common trap: Many students try to guess values like (4, 12) or (6, 6) without a system — this wastes time and risks missing cases. The real trap here is the pair (6, 6), which satisfies the reciprocal equation but violates the "distinct" constraint. Always re-read the stem.
Takeaway: Whenever you see 1/x + 1/y = 1/k with integer constraints, reach for Simon's Favorite Factoring Trick — rearrange, add a constant to both sides, and factor. It turns an open-ended search into a clean factor-pair problem every time.
Answer: D (16)
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Free gamified GMAT prep: edskore.com | 725 GMAT Focus
