Hello,
Can someone please explain this question/solution to me?
Which of the following is NOT prime? I know the answer is all 3 are not prime, but how did you arrive at that answer?
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30! + 10
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30! + 29
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30! + 63
Thank you!
Hello,
In this problem, we have 30!
This means 30! is the product of all positive integers from 1 to 30.
30! = 30×29×28×…×3×2×1
Which means: any integer from 2 to 30 is a factor of 30!
30!+10
Both 30! and 10 are divisible by 10 (and also by 2 and 5).
30! is divisible by 10 because 10 is one of the numbers in the product that makes up 30!
10 is trivially divisible by 10.
Since both parts of the sum are divisible by 10, their sum, 30!+10, must also be divisible by 10. Therefore, it is not a prime number.
30!+29
Both 30! and 29 are divisible by 29.
30! is divisible by 29 because 29 is one of the factors in its calculation.
29 is divisible by 29.
Since both terms are divisible by 29, their sum, 30!+29, is also divisible by 29. Thus, it is not a prime number.
30!+63
Both 30! and 63 are divisible by a common factor.
First, we need to find the factors of 63. 63 = 9×7 = 32×7. So, the factors of 63 are 1, 3, 7, 9, 21, and 63. Let’s see if any of these are also factors of 30!
30! is divisible by 9 (since 9 is a number in the product 1×…×9×…×30).
30! is also divisible by 7 (since 7 is a number in the product).
Since both 30! and 63 are divisible by 9 (or 7), their sum, 30!+63, must also be divisible by 9 (or 7). Therefore, it is not a prime number.
I hope this helped.
Thank you.