Home/Class 10/Maths/

Question and Answer

Show that any positive odd integer is of the form \(6q+1,6q+3\) or \( 6q+5,\) where \(q\) is some integer.
loading
settings
Speed
00:00
07:48
fullscreen
Show that any positive odd integer is of the form \(6q+1,6q+3\) or \( 6q+5,\) where \(q\) is some integer.

Answer

See solution below 
To Keep Reading This Answer, Download the App
4.6
star pngstar pngstar pngstar pngstar png
Review from Google Play
To Keep Reading This Answer, Download the App
4.6
star pngstar pngstar pngstar pngstar png
Review from Google Play

Solution

Let \(a\) be any positive odd integer.
On taking, \(b=6\) and on applying Euclid's division lemma on \(a\) and \(6\) there exist unique integers \(q\) and \(r\) such that:
\(a=6q+r\;\;\;[0\leq r< 6\;\mathrm{i.e,}\;r =0 ,1,2,3,4,5]\)
If \(r=0,\Rightarrow a=6q,\) and  \(6q = 2(3q)\) is divisible by 2 
\(\Rightarrow\) \(6q\)is even.
If \(r=1,\Rightarrow a=6q+1\) and \(6q+1\) is not divisible by 2.
If \(r=2,\Rightarrow a=6q+2\) and \(6q+2=2(3q+1)\) is divisible by 2 
\(\Rightarrow 6q+2\) is even.
If \(r=3,\Rightarrow a=6q+3,\) and \(6q+3\) is not divisible by 2.
If \(r=4,\Rightarrow a=6q+4,\) and \(6q+4=2(3q+2)\) is divisible by 2
\(\Rightarrow\) \(6q+4\) is even.
If \(r=5,\Rightarrow a=6q+5\) and \(6q+5\) is not divisible by 2.
Since, \(6q,6q+2,6q+4\) are even . So these cases will not be considered as a is an odd positive integer and the remaining cases \(6q+1,6q+3\) and \(6q+5\)are odd.
So, we can conclude that all the positive odd integers will be of the form of \(6q+1,6q+3\) or \(6q+5\)
To Keep Reading This Solution, Download the APP
4.6
star pngstar pngstar pngstar pngstar png
Review from Google Play
To Keep Reading This Solution, Download the APP
4.6
star pngstar pngstar pngstar pngstar png
Review from Google Play
Correct37
Incorrect2
Watch More Related Solutions
Use Euclid’s division lemma to show that the square of any positive integer is either of the form \(3m \;or\;3m+1\) for some integer \(m.\)
Express this number as a product of its prime factors
\(140\)
Find the LCM and HCF of \(26 \)and \(91\) and verify that LCM \(\times\) HCF \(=\)Product of two numbers
Find the LCM and HCF of the following integers by applying the prime factorsation method. 12,15 and 21
Given that HCF\((306,657)=9\),find LCM\((306,657)\).
Check whether\(6^{n}\)can end with the digit \(0\) for any natural number \(n\).
Explain why \(7\times 11\times 13+13\) and \(7\times 6\times 5\times 4\times 3\times 2\times 1+5\) are composite numbers?
There is a circular path around a sports field. Sonia takes18 minutes to drive one round of the field, while Ravi takes 12 minutes for the same. Suppose they both start at the same point and at the same time, and go in the same direction, After how many minutes will they meet again at the starting point?
Prove that \(\sqrt {5}\) is an irrational number.
Prove that\(3+ 2\sqrt {5}\)is irrational.

Load More