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An Integer Product (Posted on 2022-05-26)

P(n) is defined as an n-term product (4-2/1)*(4-2/2)*...*(4-2/n).

Prove P(n) is an integer for all natural numbers n.

See The Solution Submitted by Brian Smith

Rating: 5.0000 (1 votes)

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re(3): demonstration

| Comment 6 of 7 |

(In reply to re(2): demonstration by Kenny M)

Take n=5; 7 is prime.

(2n)! = 10*9*8*7*6*5*4*3*2

(n!)^2 = 5*4*3*2 * 5*4*3*2

Factor by factor:

10/5 = 2

9 = 9

8/(4*2) = 1

7 = 7

6/3 = 2

5/5=1

4/4 = 1

3/3=1

2/2=1

Multiply together and it's 2*9*7*2 = 252, an integer.

The 7 in the numerator did no harm at all. It would be a different story if it were in the denominator, but it's not.

This agrees with my first post for n=5.

Posted by Charlie on 2022-05-27 06:49:06

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