For any integer m greater than 1, $m denotes the product

[M7MBA]

For any integer m greater than 1, $m denotes the product of all the integers from 1 to m, inclusive. How many prime numbers are there between $7 + 2 and $7 + 10, inclusive?

(A) None
(B) One
(C) Two
(D) Three
(E) Four

The OA is A.

I don't have this PS clear. Experts, can you give me your help please? Thanks in advanced.

[Brent@GMATPrepNow]

For any integer m greater than 1, $m denotes the product of all the integers from 1 to m, inclusive. How many prime numbers are there between $7 + 2 and $7 + 10, inclusive?

(A) None
(B) One
(C) Two
(D) Three
(E) Four

$7 + 2 = (7)(6)(5)(4)(3)(2)(1) + 2
= 2[(7)(6)(5)(4)(3)(1) + 1]
We can see that $7 + 2 is a multiple of 2, which means $7 + 2 is NOT prime

$7 + 3 = (7)(6)(5)(4)(3)(2)(1) + 3
= 3[(7)(6)(5)(4)(2)(1) + 1]
We can see that $7 + 3 is a multiple of 3, which means $7 + 3 is NOT prime

$7 + 4 = (7)(6)(5)(4)(3)(2)(1) + 4
= 4[(7)(6)(5)(3)(2)(1) + 1]
We can see that $7 + 4 is a multiple of 4, which means $7 + 4 is NOT prime
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.
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$7 + 8 = (7)(6)(5)(4)(3)(2)(1) + 8
= 8[(7)(6)(5)(3)(1) + 1] [aside: notice that (4)(2) = 8, so I factored the 4 and 2 out]
We can see that $7 + 8 is a multiple of 8, which means $7 + 8 is NOT prime

$7 + 9 = (7)(6)(5)(4)(3)(2)(1) + 9
= (7)(2)(3)(5)(4)(3)(2)(1) + 9
= 9[(7)(2)(5)(4)(2)(1) + 1]
We can see that $7 + 9 is a multiple of 9, which means $7 + 9 is NOT prime

$7 + 10 = (7)(6)(5)(4)(3)(2)(1) + 10
= 10[(7)(6)(4)(3)(1) + 1]
We can see that $7 + 10 is a multiple of 10, which means $7 + 10 is NOT prime

Answer: A

Cheers,
Brent

[Brent@GMATPrepNow]

By the way, this question is a variation of 2 official GMAT questions. If you'd like more practice, try these two:

- https://www.beatthegmat.com/laymans-term ... t9968.html
- https://www.beatthegmat.com/help-on-quan ... 26500.html

Cheers,
Brent

[Jeff@TargetTestPrep]

For any integer m greater than 1, $m denotes the product of all the integers from 1 to m, inclusive. How many prime numbers are there between $7 + 2 and $7 + 10, inclusive?

(A) None
(B) One
(C) Two
(D) Three
(E) Four

We see that $m is the conventional notation of m!. Thus, the problem asks for the number of prime numbers between 7! + 2 and 7! + 10, inclusive. Let's analyze each of these numbers.

7! + 2: Since 2 divides into 7! and 2, 7! + 2 has 2 as a factor, and thus it's not a prime.

7! + 3: Since 3 divides into 7! and 3, 7! + 3 has 3 as a factor, and thus it's not a prime.

7! + 4: Since 4 divides into 7! and 4, 7! + 4 has 4 as a factor, and thus it's not a prime.

7! + 5: Since 5 divides into 7! and 5, 7! + 5 has 5 as a factor, and thus it's not a prime.

7! + 6: Since 6 divides into 7! and 6, 7! + 6 has 6 as a factor, and thus it's not a prime.

7! + 7: Since 7 divides into 7! and 7, 7! + 7 has 7 as a factor, and thus it's not a prime.

7! + 8: Since 8 divides into 7! (notice that 7! has factors 2 and 4) and 8, 7! + 8 has 8 as a factor, and thus it's not a prime.

7! + 9: Since 9 divides into 7! (notice that 7! has factors 3 and 6) and 9, 7! + 9 has 9 as a factor, and thus it's not a prime.

7! + 10: Since 10 divides into 7! (notice that 7! has factors 2 and 5) and 10, 7! + 10 has 10 as a factor, and thus it's not a prime.

Thus, none of the integers between 7! + 2 and 7! + 10, inclusive, are prime.

Answer: A