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655-705 (Hard)|   Sequences|      
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Bunuel
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Given a sequence of 58 terms; such that each term has the form P + n 15 Jun 2021, 02:15
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55% (hard)

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62% (02:08) correct 38% (02:23) wrong based on 246 sessions

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Given a sequence of 58 terms; such that each term has the form P + n where P stands for the product 2 × 3 × 5 …× 61 of all prime numbers less than or equal to 61, and ‘n’ takes, successively, the values 2, 3, 4, …, 59. If M is the number of primes appearing in this sequence, then what is the value of M?

(A) 0
(B) 16
(C) 17
(D) 57
(E) 58
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A
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Given a sequence of 58 terms; such that each term has the form P + n 15 Jun 2021, 05:08
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P is divisible by every prime from 2 through 61. All of the numbers n we'll be adding to P have at least one prime divisor between 2 and 61. So we'll always be able to factor out that prime divisor from the sum, and not one of the numbers in this sequence can be prime.
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Given a sequence of 58 terms; such that each term has the form P + n 31 Mar 2024, 07:39
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Hello Bunuel, Can you please provide the explanation to this question?­
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Given a sequence of 58 terms; such that each term has the form P + n 31 Mar 2024, 12:13
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Prince1890Sharma
Given a sequence of 58 terms; such that each term has the form P + n where P stands for the product 2 × 3 × 5 …× 61 of all prime numbers less than or equal to 61, and ‘n’ takes, successively, the values 2, 3, 4, …, 59. If M is the number of primes appearing in this sequence, then what is the value of M?

(A) 0
(B) 16
(C) 17
(D) 57
(E) 58

Hello Bunuel, Can you please provide the explanation to this question?­
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The sequence will be:

2 × 3 × 5 …× 61 + 2 = 2(1 × 3 × 5 …× 61 + 1)
2 × 3 × 5 …× 61 + 3 = 3(2 ×1 × 5 …× 61 + 1)
2 × 3 × 5 …× 61 + 4 = 2(1 × 3 × 5 …× 61 + 2)
...
2 × 3 × 5 …× 61 + 58 = 2(1 × 3 × 5 …× 61 + 29)
2 × 3 × 5 …× 61 + 59 = 59(2 × 3 × 5 …× 1 × 61 + 1)
Observe that since 2 × 3 × 5 …× 61 is the product of all the primes from 2 to 61, inclusive, from each term we were able to factor out a prime both from the first term, 2 × 3 × 5 …× 61, as well as from the second term, 2, 3, 4, ..., 59. Hence, each term will be represnted as the product of two numbers each of which is greater than 1. Hence, none of the terms is prime.

Answer: A.
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Given a sequence of 58 terms; such that each term has the form P + n 13 Oct 2025, 06:38
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