2. Show that any positive odd integer is of the form 6q +1,or 6q+3,6q+5, where q is some integer
Answers
Answer:
Step-by-step ex(i) Let x=0.
47
. Then x=0.474747....
Since two digits are repeating, multiplying both sides by 100, we get
100x=47.474747...=47+0.474747...=47+x
99x=47
x=
99
47
∴0.
47
=
99
47
(ii) Let x=0.
001
. Then x=0.001001001...
Since three digits are repeating, multiplying both sides by 1000, we get
1000x=1.001001001...=1+0.001001001...=1+x
1000x−x=1
999x=1
x=
999
1
∴0.
001
=
999
1
(iii) Let x=0.5
7
. Then x=0.57777....
Multiplying both sides by 10, we get
10x=5.7777....=5.2+0.57777...=5.2+x
9x=5.2
x=
9
5.2
x=
90
52
∴0.5
7
=
90
52
=
45
26
(iv) Let x=0.2
45
. Then x=0.2454545...
Multiplying both sides by 100, we get
100x=24.545454...=24.3+0.2454545...=24.3+x
99x=24.3
x=
99
24.3
0.2
45
=
990
243
=
110
27
(v) Let x=0.
6
. Then x=0.66666...
Multiplying both sides by 10, we get
10x=6.66666....=6+0.6666...=6+x
9x=6
x=
9
6
=
3
2
∴0.
6
=
3
2
(vi) Let x=1.
5
. Then x=1.55555...
Multiplying both sides by 10, we get
10x=15.5555...=14+1.5555...=114+x
9x=14
x=
9
14
∴1.
5
=1
9
5
So, every number with a non-terminating and recurring decimal expansion can be expressed in the form
q
p
, where p and q are integers and q not equal to zero.planation:
Using Euclid division algorithm, we know that a = bq + r, 0 ≤ r ≤ b ----(1)
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
Therefore, a = 6q or 6q + 1 or 6q + 2 or 6q + 3 or 6q + 4 or 6q + 5
6q + 0 : 6 is divisible by 2, so it is an even number.
6q + 1 : 6 is divisible by 2, but 1 is not divisible by 2 so it is an odd number.
6q + 2 : 6 is divisible by 2, and 2 is divisible by 2 so it is an even number.
6q + 3 : 6 is divisible by 2, but 3 is not divisible by 2 so it is an odd number.
6q + 4 : 6 is divisible by 2, and 4 is divisible by 2 so it is an even number.
6q + 5 : 6 is divisible by 2, but 5 is not divisible by 2 so it is an odd number.
And therefore, any odd integer can be expressed in the form 6q + 1 or 6q + 3 or 6q+5.
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