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A set of 4 distinct positive integers has a sum of 20. What is the largest of the four integers?
(1) The smallest of the four integers is 2.
(2) The four integers, in increasing order, form an arithmetic sequence.
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Correct answer
B
Four distinct positive integers, sum 20.
Statement (1): smallest = 2, so the other three distinct integers exceed 2 and sum to 18. They could be 3, 4, 11 (largest 11) or 3, 6, 9 (largest 9), so the largest is not fixed. not sufficient.
Statement (2): an arithmetic sequence a, a+d, a+2d, a+3d (with d a positive integer so the four terms are distinct) sums to 4a + 6d = 20, so 2a + 3d = 10. For a to be a positive integer, 10 − 3d must be positive and even: d = 1 gives 2a = 7 (odd, no), d = 2 gives 2a = 4 so a = 2 and the sequence is 2, 4, 6, 8 (largest 8), d = 3 gives 2a = 1 (odd, no), d = 4 gives 2a = −2 (not positive, no). So the only valid sequence is 2, 4, 6, 8, with largest 8. determinate. sufficient.
The (C) lure thinks the smallest value is needed, but the arithmetic constraint plus the fixed sum has a unique positive-integer solution, fixing the largest at 8.
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