Advertisements
Advertisements
Question
Show that every positive integer is either even or odd?
Solution
Let us assume that there exist a smallest positive integer that is neither odd nor even, say n. Since n is least positive integer which is neither even nor odd, n – 1 must be either odd or even.
Case 1: If n – 1 is even, n – 1 = 2k for some k.
But this implies n = 2k + 1
this implies n is odd.
Case 2: If n – 1 is odd, n – 1 = 2k + 1 for some k.
But this implies n = 2k + 2 (k+1)
this implies n is even.
In both ways we have a contradiction.
Thus, every positive integer is either even or odd.
APPEARS IN
RELATED QUESTIONS
Show that any positive odd integer is of the form 4q + 1 or 4q + 3, where q is some integer.
Find the HCF of the following pairs of integers and express it as a linear combination of 506 and 1155.
During a sale, colour pencils were being sold in packs of 24 each and crayons in packs of 32 each. If you want full packs of both and the same number of pencils and crayons, how many of each would you need to buy?
Using prime factorization, find the HCF and LCM of 23, 31 In case verify that HCF × LCM = product of given numbers.
Find the smallest number which when divides 28 and 32, leaving remainders 8 and 12 respectively.
Three pieces of timber 42m, 49m and 63m long have to be divided into planks of the same length. What is the greatest possible length of each plank? How many planks are formed?
Without actual division show that each of the following rational numbers is a non-terminating repeating decimal.
(i) `29/343`
Express each of the following integers as a product of its prime factors:
945
If HCF (26, 169) = 13, then LCM (26, 169) =
Show that 107 is of the form 4q +3 for any integer q
