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Given:
1/x + 1/y + 1/z = 1
Check which values x + y + z could take.
---
Initial Constraints
x, y, z are positive integers.
None can be 1 because:
1 + 1/y + 1/z > 1
So:
x, y, z ≥ 2
Also:
1/2 + 1/2 = 1
Therefore at most one variable can equal 2.
---
Test I: x + y + z = 7
Smallest possible integers ≥ 2 summing to 7:
2, 2, 3
Check:
1/2 + 1/2 + 1/3 = 4/3
Not equal to 1.
So 7 is impossible.
---
Test II: x + y + z = 9
Try:
3, 3, 3
Check:
1/3 + 1/3 + 1/3 = 1
Works.
So 9 is possible.
---
Test III: x + y + z = 11
Try:
2, 3, 6
Check:
1/2 + 1/3 + 1/6 = 1
Works.
So 11 is possible.
---
Answer
II and III only
E
---
GMAT Pattern
For:
1/x + 1/y + 1/z = 1
Memorize the only positive-integer solution sets (up to rearrangement):
(3,3,3)
(2,4,4)
(2,3,6)
Corresponding sums:
3 + 3 + 3 = 9
2 + 4 + 4 = 10
2 + 3 + 6 = 11
Therefore possible values of x + y + z are only:
9, 10, 11
Any other sum is impossible.
1/x + 1/y + 1/z = 1
Check which values x + y + z could take.
---
Initial Constraints
x, y, z are positive integers.
None can be 1 because:
1 + 1/y + 1/z > 1
So:
x, y, z ≥ 2
Also:
1/2 + 1/2 = 1
Therefore at most one variable can equal 2.
---
Test I: x + y + z = 7
Smallest possible integers ≥ 2 summing to 7:
2, 2, 3
Check:
1/2 + 1/2 + 1/3 = 4/3
Not equal to 1.
So 7 is impossible.
---
Test II: x + y + z = 9
Try:
3, 3, 3
Check:
1/3 + 1/3 + 1/3 = 1
Works.
So 9 is possible.
---
Test III: x + y + z = 11
Try:
2, 3, 6
Check:
1/2 + 1/3 + 1/6 = 1
Works.
So 11 is possible.
---
Answer
II and III only
E
---
GMAT Pattern
For:
1/x + 1/y + 1/z = 1
Memorize the only positive-integer solution sets (up to rearrangement):
(3,3,3)
(2,4,4)
(2,3,6)
Corresponding sums:
3 + 3 + 3 = 9
2 + 4 + 4 = 10
2 + 3 + 6 = 11
Therefore possible values of x + y + z are only:
9, 10, 11
Any other sum is impossible.
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