Which of the following statements is wrong?
(a) Product of two consecutive positive integers is divisible by 2
(b) Product of 3 consecutive positive integers is always divisible by 3
(c) Square of an odd positive integer is always of the form 8K + 1
(d) Square of any positive integer is always of the form 5K or 5K + 4
Answers
option A is the wrong statement brooooooooo.
1 . integer is of one of the form 2q, 2q+1. Hence n(n+1) = 2((2q+1)(q+1)), which is even. Hence n(n+1) is always even. Hence the product of two consecutive integers is always divisible by 2.
2.If n = 3p + 2, then n + 1 = 3p + 2 + 1 = 3p + 3 = 3(p + 1) is divisible by 3. So, we can say that one of the numbers among n, n + 1 and n + 2 is always divisible by 3. ⇒ n (n + 1) (n + 2) is divisible by 3. Similarly, whenever a number is divided 2, the remainder obtained is 0 or 1.
3. Any odd integer can be represented by 2m+1 for some integer m (can prove that 2m+1 is always an odd number if needed). (2m+1)2=4m2+2m+2m+1=4m2+4m+1=4m(m+1)+1 simple factorisation. m and (m+1) are consecutive integers, one of them has to be an even number, when an odd number is multiplied by an even number the result is always an even number (I will prove it to the student if needed), since m(m+1) is even it can be written in the form 2k for some integer k, now 4m(m+1)+1=4(2k)+1=8k+1 hence the square of any odd integer can be written in the form (8k+1). proof completed
4. When r = 4, we get, a = 5k + 4
Therefore, the square of any positive integer is of the form 5q or, 5q + 1 or 5q + 4 for some integer q. proved ...