show that any poitive odd integer is of the form 8m+1,8m+5,8m+3 or 8m+7 where m is some integer
pls tell with steps
Answers
Answer:
what is potitive integer in math!!!!!!?????
Step-by-step explanation:
Solution :-
We know that
Euclid's Division Lemma
For any two positive integers there exists two positive integers q and r satisfying a = bq+r , 0≤r<b
Let b = 8 , q = m
On writing it a = bq+r then
a = 8m+r , 0≤r<8 ------------(1)
The possible values of r = 0,1,2,3,4,5,6,7
I) Put r = 0 in (1) then
a = 8m+0
=> a = 8m -------------------(2)
=> a = 2(4m)
It is an even integer
ii) If r = 1 then
a = 8m +1--------------------(3)
It is an odd integer
iii) if r = 2 in (1) then
=> a = 8m +2-------------(4)
=> a = 2(4m+1)
It is an even integer.
iv) If r = 3 in (1) then
=> a = 8m +3 -----------(5)
=> a = 8m +2 +1
=> a = 2(4m+1)+1
It is an odd integer
v) If r = 4 in (1) then
a = 8m +4--------------------(6)
=> a = 2(4m+2)
It is an even integer
vi) Put r = 5 in (1) then
a = 8m+5--------------(7)
=> a =8m+4+1
=> a = 2(4m+2)+1
It is an odd integer
ii) If r = 6 in (1) then
a = 8m +6--------------------(8)
=> a = 2(4m+3)
It is an even integer
iii) if r = 7 in (1) then
=> a = 8m +7-------------(9)
=> a = 8m+6+1
=> a = 2(4m+3)+1
It is an odd integer.
From (3),(5),(7)&(9)
We conclude that
"Any poitive odd integer is of the form 8m+1,8m+5,8m+3 or 8m+7 where m is some integer.
Used formulae:-
Euclid's Division Lemma:-
For any two positive integers there exists two positive integers q and r satisfying a = bq+r , 0≤r<b.
Integers :-
Positive numbers, negative numbers and zero together called the set of integers .
Positive integers :-
1,2,3,4,5 ...are called the positive integers.
Even integer:-
The integers are in the form of 2n,where n is the positive integer called even integers.
Ex:- 2,4,6...
All multiples of 2 are even numbers.
Odd integer:-
The integers are in the form of 2n+1 ,where n is the positive integer called odd integers.
Ex:- 1,3,5...