## Suppose x is the product of all the primes less than or equal to 59. How many primes appear in the set

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### Suppose x is the product of all the primes less than or equal to 59. How many primes appear in the set

by BTGmoderatorDC » Mon Jun 21, 2021 3:44 pm

00:00

A

B

C

D

E

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Suppose x is the product of all the primes less than or equal to 59. How many primes appear in the set {x + 2, x + 3, x + 4, …, x + 59}?

A. 0
B. 17
C. 18
D. 23
E. 24

OA A

Source: Veritas Prep

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### Re: Suppose x is the product of all the primes less than or equal to 59. How many primes appear in the set

by swerve » Tue Jun 22, 2021 2:51 pm
BTGmoderatorDC wrote:
Mon Jun 21, 2021 3:44 pm
Suppose x is the product of all the primes less than or equal to 59. How many primes appear in the set {x + 2, x + 3, x + 4, …, x + 59}?

A. 0
B. 17
C. 18
D. 23
E. 24

OA A

Source: Veritas Prep
Since $$2 \cdot 3 \cdot 5 \cdot \ldots \cdot 59$$ is even, $$x$$ is even.

As $$x$$ is even, so will be $$x+2, x+4, x+6, \ldots, x+58,$$ so these numbers cannot be prime.

Now, $$x+3$$ is nothing but $$(2\cdot 3\cdot \ldots \cdot 59) + 3.$$ Rewriting this as $$\{3 \cdot (2 \cdot 5\cdot \ldots \cdot 59)\} +3.$$

One can observe here that the term in the curly brace is a multiple of $$3$$ and that term plus $$3$$ will definitely be a multiple of $$3.$$

(Multiple of $$3 + 3$$ is a multiple of $$3,$$ Multiple of $$59 + 59$$ is a multiple of $$59$$)

As $$x+3$$ is a multiple of $$3$$ it cannot be a prime.

Similarly, $$x+5$$ can be written as $$\{5 \cdot (2 \cdot 3 \cdot 7 \cdot \ldots \cdot 59)\} +5$$ which is a multiple of $$5.$$

Like wise, we can see that all the terms in the set are multiples of prime numbers.

Therefore, the correct answer is A

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