Question -
Answer -
Solution:
No, the square of any positive integer cannotbe written in the form 3m + 2 where m is a natural number
Justification:
According to Euclid’s division lemma,
A positive integer ‘a’ can be written in theform of bq + r
a = bq + r, where b, q and r are any integers,
For b = 3
a = 3(q) + r, where, r can be an integers,
For r = 0, 1, 2, 3……….
3q + 0, 3q + 1, 3q + 2, 3q + 3……. are positiveintegers,
(3q)2 = 9q² = 3(3q²) = 3m(where 3q² = m)
(3q+1)2 = (3q+1)² = 9q²+1+6q =3(3q²+2q) +1 = 3m + 1 (Where, m = 3q²+2q)
(3q+2)2 = (3q+2)² = 9q²+4+12q= 3(3q²+4q) +4 = 3m + 4 (Where, m = 3q²+2q)
(3q+3)2 = (3q+3)² = 9q²+9+18q= 3(3q²+6q) +9 = 3m + 9 (Where, m = 3q²+2q)
Hence, there is no positive integer whosesquare can be written in the form 3m + 2 where m is a natural number.