Prove that one and only one out of n, n + 2 and n + 4 is divisible by 3, where n is any positive integer.
Show that the cube of a positive integer of the form 6q + r, q is an integer and r = 0, 1, 2, 3, 4, 5 is also of the form 6m + r.
Show that the square of an odd positive integer can be of the form 6q + 1 or 6q + 3 for some integer q.
Write the denominator of the rational number 257/ 5000 in the form 2m × 5n, where m, n are non-negative integers. Hence, write its decimal expansion, without actual division.
On a morning walk, three persons step off together and their steps measure 40 cm, 42 cm and 45 cm, respectively. What is the minimum distance each should walk so that each can cover the same distance in complete steps?
Using Euclid’s division algorithm, find the largest number that divides 1251, 9377 and 15628 leaving remainders 1, 2 and 3, respectively.
Prove that if x and y are both odd positive integers, then x2 + y2 is even but not divisible by 4.
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