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- ray_serrano
- Course Students
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NP Chapter 1 Page 39 Exercise 30
Can you explain why the fact that 6! and 41 do not share any prime factors makes 6!+41 a possible prime.
- JonathanSchneider
- ManhattanGMAT Staff
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Re: NP Chapter 1 Page 39 Exercise 30
Great question.
Imagine the following algebraic form:
ax + b
Now, if a, x, and b are all positive integers, does the sum above equal a prime number? Well, how might we figure that out? We can only determine that something is NOT prime by showing that it has a factor other than 1 and itself. So, we might start determining if either a or x shares a factor with b. If so, then the sum is not prime. For example, let us imagine that a = 2, x = 3, and b = 4. The form would then be:
(2)(3) + 4 = 10
This is not prime. We could have proven that the number would be none-prime, of course, because a and b each share a 2 as a factor. In other words, we could have pulled the 2 out, as such:
2(3 + 2)
Note that the above example shows that the common factor of a and b has been pulled out. Whatever the sum is of the remaining pieces, the overall product will be non-prime.
But what if neither a nor x shared a common factor with b? Then we couldn't pull anything out, so we would not be able to prove that the sum was non-prime. For example, the form might be:
(2)(3) + (5)
Neither 2 nor 3 shares a common factor with 5; thus, we cannot pull out any common factor to show that the overall sum is prime.
Of course, we could have arrived at a prime sum this way. Consider:
(2)(3) + 19
There are no shared factors above, yet the sum equals 25, which is non-prime. The idea here is that we can demonstrate when a sum will be non-prime, but we cannot demonstrate when it will be prime (until we actually solve for the sum). (For that reason, the problem that you cite is best done as a process of elimination.)
The same process holds, of course, for more complex forms. Consider:
abcdef + g
If ANY of a, b, c, d, e, or f shares a common factor (greater than 1) with g, then the sum will be non-prime. This is because we could pull that factor out, to show the sum as a product.
Now, 6! equals (6)(5)(4)(3)(2)(1). 41 is itself prime. Thus, there are no shared factors between 6! and 41. As a result, we cannot easily show that this sum is non-prime. We have not proven that it IS prime, but we do not know that it is not.
Imagine the following algebraic form:
ax + b
Now, if a, x, and b are all positive integers, does the sum above equal a prime number? Well, how might we figure that out? We can only determine that something is NOT prime by showing that it has a factor other than 1 and itself. So, we might start determining if either a or x shares a factor with b. If so, then the sum is not prime. For example, let us imagine that a = 2, x = 3, and b = 4. The form would then be:
(2)(3) + 4 = 10
This is not prime. We could have proven that the number would be none-prime, of course, because a and b each share a 2 as a factor. In other words, we could have pulled the 2 out, as such:
2(3 + 2)
Note that the above example shows that the common factor of a and b has been pulled out. Whatever the sum is of the remaining pieces, the overall product will be non-prime.
But what if neither a nor x shared a common factor with b? Then we couldn't pull anything out, so we would not be able to prove that the sum was non-prime. For example, the form might be:
(2)(3) + (5)
Neither 2 nor 3 shares a common factor with 5; thus, we cannot pull out any common factor to show that the overall sum is prime.
Of course, we could have arrived at a prime sum this way. Consider:
(2)(3) + 19
There are no shared factors above, yet the sum equals 25, which is non-prime. The idea here is that we can demonstrate when a sum will be non-prime, but we cannot demonstrate when it will be prime (until we actually solve for the sum). (For that reason, the problem that you cite is best done as a process of elimination.)
The same process holds, of course, for more complex forms. Consider:
abcdef + g
If ANY of a, b, c, d, e, or f shares a common factor (greater than 1) with g, then the sum will be non-prime. This is because we could pull that factor out, to show the sum as a product.
Now, 6! equals (6)(5)(4)(3)(2)(1). 41 is itself prime. Thus, there are no shared factors between 6! and 41. As a result, we cannot easily show that this sum is non-prime. We have not proven that it IS prime, but we do not know that it is not.
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