Show that one and only one out of n,n+2,n+4 is divisible by 3, where n is any positive integer.
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Show that any positive odd integer is of the form 6q+1,6q+3 & 6q+5 where q is some integer.
Solution: Let a be any positive integer and b = 6. Then, by Euclid’s algorithm, a = 6q + r for some integer q ≥ 0, and r = 0, 1, 2, 3,...
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Show that any positive odd integer is of the form 6q + 1, or 6q + 3, or 6q + 5, where q is some integer.
Solution: Let a be any positive integer and b = 6. Then, by Euclid’s algorithm, a = 6q + r for some integer q ≥ 0 and r = 0,1,2,3,4,5 because 0 ≤ r ≤ 6. So, a...
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Show that cube of any positive integer is of the form 4m, 4m + 1 or 4m + 3, for some integer m.
How to assume value of “b”? Here it is assumed as a=4q+0,1,2,3. But in some similar problems value of b is taken as 3. So what is the criteria to assume...
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Show that cube of any positive integer is of the form 4m, 4m + 1 or 4m + 3, for some integer m.
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Show that the square of any positive integer cannot be of the form 5q + 2 or 5q + 3 for any integer q.
Solution: Let a be the positive integer and b = 5. Then, by Euclid’s algorithm, a = 5m + r for some integer m ≥ 0 and r = 0, 1, 2,...
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Show that the square of any positive integer cannot be of the form 6m + 2 or 6m + 5 for any integer m.
Solution: Let a be the positive integer and b = 6. Then, by Euclid’s algorithm, a = 6q + r for some integer q ≥ 0 and r = 0, 1, 2,...
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Prove that one and only one out of n, n + 2 and n + 4 is divisible by 3, where n is any positive integer.
Solution: By Euclid’s division lemma, we have a = bq + r; 0 ≤ r < b For a = n and b = 3, we have n = 3q + r, …(i) where q is...
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Show that one and only one out of n, n + 4, n + 8, n + 12 and n + 16 is divisible by 5, where n is any positive integer.
Solution: By Euclid’s division lemma, we have a = bq + r; 0 ≤ r < b For a = n and b = 5, we have n = 5q + r …(i) Where q is...
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n^2 - 1 is divisible by 8, if n is (A) an integer (B) a natural number (C) an odd integer (D) an even integer
n2 - 1 is divisible by 8, if n is an odd integer
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Show that one and only one out of n,n+2,n+4 is divisible by 3, where n is any positive integer.
We applied Euclid Division algorithm on n and 3. a = bq +r on putting a = n and b = 3 n = 3q +r , 0<r<3 i.e n = 3q --------...
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