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4. Show that any positive odd integer is of the form 6q +1,6q+ 3 or 6q+5, where q is some
integer.
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(NCERT)
4. Show that any positive odd integer is of the form 6q +1,6q+ 3 or 6q+5, where q is some
integer.
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Mathematics
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2021-08-03T13:16:02+00:00
2021-08-03T13:16:02+00:00 2 Answers
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Answers ( )
Answer:
According to Euclid’s division lemma
a = bq + r
a = 6q + r………………….(1)
where, (0 ≤ r < 6)
So r can be either 0, 1, 2, 3, 4 and 5.
Case 1:
If r = 1, then equation (1) becomes
a = 6q + 1
The Above equation will be always as an odd integer.
Case 2:
If r = 3, then equation (1) becomes
a = 6q + 3
The Above equation will be always as an odd integer.
Case 3:
If r = 5, then equation (1) becomes
a = 6q + 5
The above equation will be always as an odd integer.
∴ Any odd integer is of the form 6q + 1 or 6q + 3 or 6q + 5.
Hence proved.
Step-by-step explanation:
Step-by-step explanation:
We know that,
a = bq+r. {by Euclid’s division lemma}
let the first integer be 6q,
second be 6q+1
third be 6q+2
fourth be 6q+3
fifth be 6q+4
sixth be 6q+5
From above we can see that second, fourth, and sixth are odd positive integers and can be written in the form 6q +1,6q+ 3 or 6q+5, while first, third,fifth cannot.
Hence proved.