Show that the square of any positive integer cannot be of the form 6m + 2 or 6m + 5 for any integer m.
Show that the square of any positive integer cannot be of the form 5q + 2 or 5q + 3 for any integer q.
Show that cube of any positive integer is of the form 4m, 4m + 1 or 4m + 3, for some integer m.
Show that the square of any positive integer is either of the form 4q or 4q + 1 for some integer q.
Show that the square of an odd positive integer is of the form 8m + 1, for some whole number m.
Using Euclid’s division algorithm, find which of the following pairs of numbers are co-prime:
(i) 231, 396 (ii) 847, 2160
A rational number in its decimal expansion is 327.7081. What can you say about the prime factors of q, when this number is expressed in the form p/q ? Give reasons.
Without actually performing the long division, find if 987/ 10500 will have terminating or non-terminating (repeating) decimal expansion. Give reasons for your answer.
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