1
Question
Every positive odd integer is of the form 2q+1, where q is some integer.
Every positive odd integer is of the form 2q+1, where q is some integer.
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Solution
The correct option is A True
Let a be any positive integer and b = 2.
By Euclid's division lemma, there exist integers q and r such that:
a = 2q + r, where 0 ≤ r < 2
⇒ 0 ≤ r ≤ 1
⇒ r = 0 or 1
When r = 0, a = 2q which is completely divisible by 2. Hence it is a positive even integer.
When r = 1, a = 2q + 1 which is gives a remainder 1 when divided by 2. Hence it is an odd integer.
Therefore, every positive odd integer is of the form 2q + 1.
True
Let a be any positive integer and b = 2.
By Euclid's division lemma, there exist integers q and r such that:
a = 2q + r, where 0 ≤ r < 2
⇒ 0 ≤ r ≤ 1
⇒ r = 0 or 1
When r = 0, a = 2q which is completely divisible by 2. Hence it is a positive even integer.
When r = 1, a = 2q + 1 which is gives a remainder 1 when divided by 2. Hence it is an odd integer.
Therefore, every positive odd integer is of the form 2q + 1.
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