Mathematics Question on Some More Interesting Patterns

Question

The following numbers are obviously not perfect squares. Give reason.

1057
23453
7928
222222
64000
89722
222000
505050

Accepted Answer

The square of numbers may end with any one of the digits $0, 1, 5, 6$ , or $9$ . Also, a perfect square has even number of zeroes at the end of it.
(i) $1057$ has its unit place digit as $7$ .
Therefore, it cannot be a perfect square.

(ii) $23453$ has its unit place digit as $3$ .
Therefore, it cannot be a perfect square.

(iii) $7928$ has its unit place digit as $8$ .
Therefore, it cannot be a perfect square.

(iv) $222222$ has its unit place digit as $2$ .
Therefore, it cannot be a perfect square.

(v) $64000$ has three zeros at the end of it.
However, since a perfect square cannot end with odd number of zeroes, it is not a perfect square.

(vi) $89722$ has its unit place digit as $2$ .
Therefore, it cannot be a perfect square.

(vii) $222000$ has three zeroes at the end of it.
However, since a perfect square cannot end with odd number of zeroes, it is not a perfect square.

(viii) $505050$ has one zero at the end of it.
However, since a perfect square cannot end with odd number of zeroes, it is not a perfect square.