CameraIcon
CameraIcon
SearchIcon
MyQuestionIcon
MyQuestionIcon
1
You visited us 1 times! Enjoying our articles? Unlock Full Access!
Question

Use Euclid's division lemma to show that the cube of any positive integer is of the form 9m,9m+1or9m+8.


Open in App
Solution

Step 1: Finding the value of r.

Let us consider two positive numbers a and b where b=3

We know that According to Euclid’s Division Lemma

a=bq+r{ condition for ris(0r<b)}

a=3q+r(i)b=3

so r is an integer which lies in between 0and3

Hence r can be either 0,1or2.

Step 2 - When r=0, the equation 1 becomes

a=3q

Now, cubing both the sides, we get

a3=3q3a3=27q3a3=93q3a3=9mWhere3q3=m

Step 3:- When r=1, the equation 1 becomes

a=3q+1

Now, cubing both the sides, we get

a3=3q+13a3=3q3+13+33q21+33q12a3=27q3+1+27q2+9qa3=27q3+27q2+9q+1a3=93q3+3q2+q+1a3=9m+1Where3q3+3q2+q=m

Step 4- When r=2, the equation 1 becomes

a=3q+2

Now, cubing both the sides, we get

a3=3q+23a3=3q3+23+33q22+33q22a3=27q3+8+54q2+36qa3=27q3+54q2+36q+8a3=93q3+6q2+4q+8a3=9m+8Where3q3+6q2+4q=m

So a can be any of the form 9mor9m+1or,9m+8

Hence proved.


flag
Suggest Corrections
thumbs-up
0
Join BYJU'S Learning Program
similar_icon
Related Videos
thumbnail
lock
Why Divisibility Rules?
MATHEMATICS
Watch in App
Join BYJU'S Learning Program
CrossIcon