If cube of 5 can be cwritten in the form amon
amt or amt8, then in (mis positive Integer) =
Answers
Answer:
As per Euclid's Division Lemma
If a and b are 2 positive integers, then
a=bq+r
Where 0≤r≤b
Let b=8,
Therefore,
a=8q+r
r=0,1,3,5
Case I:- r=0
Therefore,
a=8q
Cubing both sides, we get
(a)
3
=(8q)
3
a
3
=512q
3
⇒a
3
=8×(64q
3
)
Here m=64q
3
Case II:- r=1
Therefore,
a=8q+1
Cubing both sides, we get
(a)
3
=(8q+1)
3
a
3
=(8q)
3
+(1)
3
+3(8q)
2
(1)+3(8q)(1)
2
a
3
=512q
3
+1+192q
2
+24q
⇒a
3
=8(64q
3
+24q
2
+3q)+1
Here m=(64q
3
+24q
2
+3q)
Case III:- r=3
Therefore,
a=8q+3
Cubing both sides, we get
(a)
3
=(8q+3)
3
a
3
=(8q)
3
+(3)
3
+3(8q)
2
(3)+3(8q)(3)
2
a
3
=512q
3
+27+576q
2
+216q
⇒a
3
=8(64q
3
+72q
2
+27q+3)+3
Here m=(64q
3
+72q
2
+27q+3)
Case IV:- r=5
Therefore,
a=8q+5
Cubing both sides, we get
(a)
3
=(8q+5)
3
a
3
=(8q)
3
+(5)
3
+3(8q)
2
(5)+3(8q)(5)
2
a
3
=512q
3
+125+960q
2
+600q
⇒a
3
=8(64q
3
+120q
2
+75q+15)+5
Here m=(64q
3
+120q
2
+75q+15)
Case V:- r=7
Therefore,
a=8q+7
Cubing both sides, we get
(a)
3
=(8q+7)
3
a
3
=(8q)
3
+(7)
3
+3(8q)
2
(7)+3(8q)(7)
2
a
3
=512q
3
+343+1344q
2
+1176q
⇒a
3
=8(64q
3
+168q
2
+147q+42)+7
Here m=(64q
3
+168q
2
+147q+42)
Thus cube of any positive number can be expressed as 8m or 8m+1 or 8m+3 or 8m+5 or 8m+7.
Hence proved.