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12 Jan 2026, 05:50
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Each of the two sets consists of five consecutive prime numbers, and the sets have exactly two primes in common. If the range of the set with the smaller sum of terms is odd, what is the range of the other set?
A. 3
B. 8
C. 10
D. 12
E. 17
A. 3
B. 8
C. 10
D. 12
E. 17
12 Jan 2026, 05:50
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Official Solution:
Each of the two sets consists of five consecutive prime numbers, and the sets have exactly two primes in common. If the range of the set with the smaller sum of terms is odd, what is the range of the other set?
A. 3
B. 8
C. 10
D. 12
E. 17
Two sets of five consecutive primes that share exactly two must be:
Smaller set: \(\{p_1, p_2, p_3, p_4, p_5\}\)
Larger set: \(\{p_4, p_5, p_6, p_7, p_8\}\)
For the smaller set to have an odd range, \(p_5 - p_1\) must be odd, so \(p_5\) must be odd and \(p_1\) must be even. All primes except 2 are odd, so this happens only if \(p_1 = 2\). That forces the smaller set to be \(\{2, 3, 5, 7, 11\}\), whose range is 9.
The overlapping larger set is then \(\{7, 11, 13, 17, 19\}\), and its range is 12.
Answer: D
Each of the two sets consists of five consecutive prime numbers, and the sets have exactly two primes in common. If the range of the set with the smaller sum of terms is odd, what is the range of the other set?
A. 3
B. 8
C. 10
D. 12
E. 17
Two sets of five consecutive primes that share exactly two must be:
Smaller set: \(\{p_1, p_2, p_3, p_4, p_5\}\)
Larger set: \(\{p_4, p_5, p_6, p_7, p_8\}\)
For the smaller set to have an odd range, \(p_5 - p_1\) must be odd, so \(p_5\) must be odd and \(p_1\) must be even. All primes except 2 are odd, so this happens only if \(p_1 = 2\). That forces the smaller set to be \(\{2, 3, 5, 7, 11\}\), whose range is 9.
The overlapping larger set is then \(\{7, 11, 13, 17, 19\}\), and its range is 12.
Answer: D
