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# For any integer m greater than 1, \$m denotes the

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BSchool Forum Moderator
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For any integer m greater than 1, \$m denotes the [#permalink]

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13 Aug 2017, 12:15
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Difficulty:

65% (hard)

Question Stats:

57% (01:24) correct 43% (01:39) wrong based on 102 sessions

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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

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Re: For any integer m greater than 1, \$m denotes the [#permalink]

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13 Aug 2017, 14:53
gmatexam439 wrote:
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 = 7*6*5*4*3*2*1=5040.
So we need to check all numbers between 5042 and 5050.
All even numbers are divisable by 2.
So what about 5043, 5045, 5047, 5049.
5045 is divisable by 5.
5043 and 5049 are divisable by 3, because the sum of all digits is divisable by 3.
And the last one - 5047. We can easily find that it is divisable by 7.
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Joined: 18 Aug 2017
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Re: For any integer m greater than 1, \$m denotes the [#permalink]

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22 Aug 2017, 07:44
\$m is m!, so between 7! + 2 and 7! + 10 how many prime number are there?

All of prime numbers except 2 are odd, 7! is even so only 7! + 3, 5, 7, or 9 can make an odd number.

Prime number is divisible by 1 and itself, examine the contenders:
- 7! divisible by 3, so adding 3 or 9 will result in a number divisible by 3.
- 7! is divisible by 5 and 7 also, so adding 5 or 7 will result in a number divisible by 5 or 7.

In sum, A- None.
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Joined: 02 Aug 2009
Posts: 6225
Re: For any integer m greater than 1, \$m denotes the [#permalink]

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22 Aug 2017, 08:00
gmatexam439 wrote:
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

hi..

\$7 is nothing but 7!..
LOGIC why none is prime
so we are looking at PRIMES between 7!+2 and 7!+10..

7!= 1*2*3*4*5*6*7, so 7! is multiple of 1,2,3,4,5,6,7,8(2*5),9(3*6),10(2*5)
so when you add any of these number starting from 2 till 10 to 7! the RESULTING SUM will be a MULTIPLE of that number 2,3,..or 10..
so NONE is PRIME

A
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1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html

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Re: For any integer m greater than 1, \$m denotes the [#permalink]

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24 Aug 2017, 16:07
gmatexam439 wrote:
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.

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Re: For any integer m greater than 1, \$m denotes the   [#permalink] 24 Aug 2017, 16:07
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