33).Using Euclid’s division lemma, if the cube of any positive integer is divided by 9 thenthe possible remainders are(1) (2) (3)(4)
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Secondary School Math 5 points
Use euclids division lemma to show that the cube of any positive integer is of the form 9m, 9m+1 or 9m+8 I want clear and detailed answer steps and no spamming please
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Shubhendu8898
Shubhendu8898 Genius
According to Euclid’s Division Lemma
Let take a as any positive integer and b = 9.
Then using Euclid’s algorithm we get a = 9q + r
here r is remainder and value of q is more than or equal to 0 and r = 0, 1, 2, 3, 4, 5 , 6 , 7 , 8,
because 0 ≤r < b and the value of b is 9
Sp possible forms will 9q, 9q+1, 9q+2,9q+3,9q+4,9q+5,9q+6,9q+7 and 9q+8
to get the cube of these values use the formula
(a+b)³ = a³ + 3a²b+ 3ab² + b³
In this formula value of a is always 9q
So plug the value we get
(9q+b)³ = 729q³ + 243q²b + 27qb² + b³
Now divide by 9 we get quotient = 81q³ + 27q²b + 3qb² and remainder is b³
So we have to consider the value of b³
b = 0 we get 9m+0 = 9m
b = 1 then 1³ = 1 so we get 9m +1
b = 2 then 2³ = 8 so we get 9m + 8
b = 3 then 3³ = 27 and it is divisible by 9 so we get 9m
b = 4 then 4³ =64 divide by 9 we get 1 as remainder so we get 9m +1
b=5 then 5³=125 divide by 9 we get 8 as remainder so we get 9m+8
b=6 then 6³=216 divide by 9 no remainder there so we get 9m
b=7 then 7³ = 343 divide by 9 we get 1 as remainder so we get 9m+1
b=8 then 8³ = 512 divide by 9 we get 8 as remainder so we get 9m+8
So all values are in form of 9m , 9m+1 or 9m+8
Answer:
ANSWER:0,1,8
I hope is this is a correct answer