If a and b are two odd positive integers such that ab, then prove that one of the two numbers (𝑎+𝑏)/2 and (𝑎−𝑏)/2is odd and the other is even.Watch Solution
Prove that the product of two consecutive positive integers is divisible by 2. Watch Solution
Prove that the product of three consecutive positive integer is divisible by 6. Watch Solution
For any positive integer n , prove that n^3 − n divisible by 6. Watch Solution
Prove that if a positive integer is of the form 6q + 5, then it is of the form 3q + 2 for some integer q, but not conversely. Watch Solution
Prove that the square of any positive integer of the form 5q + 1 is of the same form. Watch Solution
Prove that the square of any positive integer is of the form 3m or, 3m + 1 but not of the form 3m +2. Watch Solution
Prove that the square of any positive integer is of the form 4q or 4q + 1 for some integer q. Watch Solution
Prove that the square of any positive integer is of the form 5q, 5q + 1, 5q + 4 for some integer q. Watch Solution
Show that the square of an odd positive integer is of the form 8q + 1, for some integer q. Watch Solution
Show that any positive odd integer is of the form 6q + 1 or, 6q + 3 or, 6q + 5, where q is some integer. Watch Solution
Prove that one of every three consecutive positive integers is divisible by 3. Watch Solution
Show that the square of any positive integer cannot be of the form 6q+2 or 6q+5 for any integer m. Watch Solution
Show that the cube of a positive integer is of the form 6q+r, where q is an integer and r= 0, 1,2,3,4,5. Watch Solution
Show that one and only one out of n, n+4,n+8,n+12and n+16 is divisible by 5, where n is any positive integer. Watch Solution
Show that the square of an odd positive integer can be of the form 6q +1 or 6q +3 for some integer q. Watch Solution
A positive integer is of the form 3q +1, q being a natural number . can you write its square in any form other than 3m +1, 3m or 3m+2 for some integer m ?justify your answer. Watch Solution
Show that the square of any positive integer cannot be of the form 3m +2, where m is a natural number. Watch Solution
