2. Show that any positive odd integer is of the form 6q+1, or 6q+3, or 6q +5, where q is
some integer.
Answers
Answer:
Let a be the positive odd integer which when divided by 6 gives q as quotient and r as remainder.
according to Euclid's division lemma
a=bq+r
a=6q+r
where , a=0,1,2,3,4,5
then,
a=6q
or
a=6q+1
or
a=6q+2
or
a=6q+3
or
a=6q+4
or
a=6q+5
but here,
a=6q+1 & a=6q+3 & a=6q+5 are odd.
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Answer:
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.
Therefore, a = 6q or 6q + 1 or 6q + 2 or 6q + 3 or 6q + 4 or 6q + 5
Also, 6q + 1 = 2 × 3q + 1 = 2k1 + 1, where k1 is a positive integer
6q + 3 = (6q + 2) + 1 = 2 (3q + 1) + 1 = 2k2 + 1, where k2 is an integer
6q + 5 = (6q + 4) + 1 = 2 (3q + 2) + 1 = 2k3 + 1, where k3 is an integer
Clearly,
6q + 1, 6q + 3, 6q + 5 are of the form 2k + 1, where k is an integer.
Therefore, 6q + 1, 6q + 3, 6q + 5 are not exactly divisible by 2.
Hence, these expressions of numbers are odd numbers.
And therefore, any odd integer can be expressed in the form 6q + 1, or
6q + 3, or 6q + 5