Problem Solving

AAPL wrote:For an integer n greater than 1, n* denotes the product of all the integers from 1 to n inclusive. How many prime numbers are there between 7*+2 and 7*+7, inclusive?

A. 0
B. 1
C. 2
D. 3
E. 4

7* = 7! = (7)(6)(5)(4)(3)(2)(1)

So, 7* + 2 = (7)(6)(5)(4)(3)(2)(1) + 2
= 2[(7)(6)(5)(4)(3)(1) + 1]
= some multiple of 2
So, 7* + 2 is NOT a prime number

7* + 3 = (7)(6)(5)(4)(3)(2)(1) + 3
= 3[(7)(6)(5)(4)(2)(1) + 1]
= some multiple of 3
So, 7* + 3 is NOT a prime number

7* + 4 = (7)(6)(5)(4)(3)(2)(1) + 4
= 4[(7)(6)(5)(3)(2)(1) + 1]
= some multiple of 4
So, 7* + 4 is NOT a prime number
.
.
.
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7* + 7 = (7)(6)(5)(4)(3)(2)(1) + 7
= 7[(6)(5)(4)(3)(2)(1) + 1]
= some multiple of 7
So, 7* + 7 is NOT a prime number

ASIDE: You can assume that I was able to perform the same steps with 7* + 5 and 7* + 6

Answer: A

Cheers,
Brent

AAPL wrote:For an integer n greater than 1, n* denotes the product of all the integers from 1 to n inclusive. How many prime numbers are there between 7*+2 and 7*+7, inclusive?

A. 0
B. 1
C. 2
D. 3
E. 4

We can see that n* actually means n!. So, 7*+ 2 = 7! + 2 and 7* + 7 = 7! + 7. Therefore, we need to determine the number of prime numbers between 7! + 2 and 7! + 7, inclusive.

The numbers between 7! + 2 and 7! + 7, inclusive, are:

7! + 2, 7! + 3, 7! + 4, 7! + 5, 7! + 6 and 7! + 7

Let's see which one(s) of these numbers are prime:

7! + 2 is divisible by 2 (since 2 divides 7! and 2 divides 2)

7! + 3 is divisible by 3 (since 3 divides 7! and 3 divides 3)

7! + 4 is divisible by 4 (since 4 divides 7! and 4 divides 4)

7! + 5 is divisible by 5 (since 5 divides 7! and 5 divides 5)

7! + 6 is divisible by 6 (since 6 divides 7! and 6 divides 6)

7! + 7 is divisible by 7 (since 7 divides 7! and 7 divides 7)

Since each of these numbers is divisible by a number other than 1 and itself, none of these numbers is a prime.

Answer: A