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There are four natural numbers which are not equal to each other. The difference between the maximum number and the minimum number is equal to 4. The product of the minimum number and the maximum number is an odd number, and the sum of these four numbers is the smallest two odd numbers. What is the product of these four numbers?
Because the sum of the four numbers is the smallest two odd numbers, the smallest two odd numbers are 11, and the product of the smallest decimal and the largest number is an odd number, which means that the two numbers are odd numbers, and the difference is 4. Therefore, the four unequal natural numbers are 1, 2, 3 and 5. The product is: 1 × 2 × 3 × 5 = 30; a: the product of the four numbers is 30
Indefinite equation: prove that the product of four consecutive positive integers cannot be a complete square number
Let these four positive integers be n, N + 1, N + 2 and N + 3 respectively,
Then n (n + 1) (n + 2) (n + 3) = [n (n + 3)] [(n + 1) (n + 2)] (commutative order)
=(n ^ 2 + 3n) (n ^ 2 + 3N + 2) (expand separately)
=(n ^ 2 + 3n) ^ 2 + 2 (n ^ 2 + 3n)
=(n^2+3n)^2+2(n^2+3n)+1-1
=(n ^ 2 + 3N + 1) ^ 2-1 (complete square formula)
The product of four consecutive positive integers is a perfect square minus one, which is certainly not a perfect square
The product of any four continuous natural numbers plus 1 must be the square of a positive integer
If yes, please explain the reason!
Yes, (n-1) n (n + 1) (n + 2) + 1 = (n ^ 2 + n-1) ^ 2
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