Question

MATHEMATICS
Real Numbers:
1. Show that every positive even integer is of the form 2m, and that every positive odd integer is of the form 2m
+1, where m is some integer.
2. Show that any positive odd integer is of the form 6m + 1, or 6m + 3, or 6m + 5, where m is some integer.
3. Explain why 7 * 11 * 13+ 13 and 7 x 6 x 5 x 4 * 3 * 2 x 1 + 5 are composite numbers.
4. Show that any positive integer is of the form 32 or 39 + 1 or 3q + 2 for some integer q.
5. Show that 5-13 is irrational.
3 marks questions
6. Check whether 6 can end with the digit 0, for any natural number r.
7. Prove that one of every three consecutive positive integers is divisible by 3.
8. Prove that n(n-1) is divisible by 2 for every positive integer n.
9. Use Euclid division lemma to show that cube of any positive integer is either of the form 9m, 9m + 1, or 9m
+8.
10. If d is the HCF of 45 and 27, find x & y satisfying d=27x +45y. (Ans d=9, x=2, y=-1).
11. Prove that if x and y are both odd positive integers, then x²+ y is even but not divisible by 4.
12. Prove that one and only one out of n, n + 2 and n + 4 is divisible by 3, where n is any positive integer.​

Answer 1

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4th answer is as follows