Question

Sir/Mam in Real Numbers chapter in the question" Use Euclid’s division lemma to show that the cube of any positive integer is of the form

9m, 9m + 1 or 9m +8" why we take 3 as the value.​

Answer 1

Answer:

■●■In proving this question we take 3 as[ (b=3) in bq+r ] the value because:

◇taking b=9 will be too long to solve because we will have to solve all 9 cases where r=0,1,2,3,4,5,6,7,8.

◇ cube of expressions of the form

3q,3q+1,3q+2 are easily divided by 9 leaving simple remainders.

For example:

♤(3q)^3=27q^3= 9(3q^3) = 9m

♤(3q+1)^3 = 27q^3 +1 +27q^2+ 9q

=9(3q^3 +3q^2 +q) +1

=9m+1

♤ (3q+2)^3=27q^3 +8 +54 q^2 +36q

=9(3q^3 + 6q^2+4q) +8

= 9m+8.

Ask for further explanation; if any.