Question

Find the tens place of the number $7^{281}\times 3^{264}$.

• A. 2
• B. 4
• C. 6
• D. 8

Answer 1

The correct option is C

6

To find the tens place, we need to find the remainder when this number is divided by 100.
$\frac{7^{281}}{100}\mid r$$=\frac{\left(\right(7^4{)}^{70}\times 7^1)}{100}\mid r$$=\frac{\left(\right(..49\times ..49{)}^{70}\times 7^1)}{100}\mid r=\frac{\left(\right(..01{)}^{70}\times 07)}{100}\mid r=..07$
Similarly, $\frac{3^{264}}{100}\mid r$$=\frac{(3^4{)}^{66}}{100}\mid r$$=\frac{(3^2\times 3^2{)}^{66}}{100}\mid r$$=\frac{(09\times 09{)}^{66}}{100}\mid r$$=\frac{{81}^{66}}{100}\mid r=..81$
(66 is of the form 5n+1, and the tens digit of ${81}^n$ repeats after a cycle of 5)

Answer is the tens digit of $07\times 81=..6.$