use division algorithm show that the cube of any positive integers is of the form 7m or 7m+1 or 7m+6
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Answer:
Step-by-step explanation:
Let ‘n’ is any positive integer, then it is of the form of 3q or, 3q + 1 or, 3q + 2
If,
n
=
3
q
By squaring both sides we get
n
2
=
(
3
q
)
2
=
9
q
2
=
3
(
3
q
2
)
Or,
n
2
=
3
m
, where
m
=
3
q
2
By squaring both sides we get
n
2
=
(
3
q
+
1
)
2
=
9
q
2
+
6
q
+
1
=
3
(
3
q
2
+
2
q
)
+
1
Or,
n
2
=
3
m
+
1
Where
m
=
(
3
q
2
+
2
q
)
If,
n
=
3
q
+
2
Then b squaring both sides we get
n
2
=
(
3
q
+
2
)
2
=
9
q
2
+
12
q
+
4
=
9
q
2
+
12
q
+
3
+
1
=
3
(
q
2
+
4
q
+
1
)
+
1
Or,
n
2
=
3
m
+
1
Where,
m
=
q
2
+
4
q
+
1
Therefore, square of any positive integer is either of the form of 3m or 3m + 1
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