Question

Show that the square of an odd positive integer is of the form 8m + 1, for
some whole number
plzz anwer fast with explaination​

Answer 1

1. Question⇒  Show that the square of an odd positive integer is of the form 8m + 1, for  some whole number

Step-by-step explanation:

• According to Euclid division lemma , a = bq + r where 0 ≤ r < b
• Here we assume b = 8 and r ∈ [1, 7 ] means r = 1, 2, 3, .....7
• Then, a = 8q + r
• Case 1 :- when r = 1 , a = 8q + 1
• squaring both sides,
• a² = (8q + 1)² = 64²q² + 16q + 1 = 8(8q² + 2q) + 1
• = 8m + 1 , where m = 8q² + 2q
• case 2 :- when r = 2 , a = 8q + 2
• squaring both sides,
• a² = (8q + 2)² = 64q² + 32q + 4 ≠ 8m +1 [ means when r is an even number it is not in the form of 8m + 1 ]
• Case 3 :- when r = 3 , a = 8q + 3
• squaring both sides,
• a² = (8q + 3)² = 64q² + 48q + 9 = 8(8q² + 6q + 1) + 1
• = 8m + 1 , where m = 8q² + 6q + 1
• You can see that at every odd values of r square of a is in the form of 8m +1
• But at every even Values of r square of a isn't in the form of 8m +1 .
• Also we know, a = 8q +1 , 8q +3 , 8q + 5 , 9q +7 are not divisible by 2 means these all numbers are odd numbers
• Hence , it is clear that square of an odd positive is in form of 8m +1