Question

The unit digit of the expression,
${{125}^{813}}\times {{553}^{3703}}\times {{4532}^{828}}$
(a)4
(b)2
(c)0
(d)5

Answer 1

Hint: Take each term separately. Take the unit’s place of each term raised to the power first. Find the power of 5, 3 and 2 when raised to the unit's place. Find the remainder of 813, 3703 and 828 which gives the raised power. Hence, multiply the unit’s place found in these 3 terms and find the unit’s place of the product.

Complete step-by-step answer:
We need to find the unit digit of given expression, which is
${{125}^{813}}\times {{553}^{3703}}\times {{4532}^{828}}$
First let us consider the term ${{125}^{813}}$ . Here 5 is in the unit place of 125 and we know that any power of 5 will give 5 in its unit’s place. Example: ${{5}^{2}}=25,{{5}^{3}}=125,{{5}^{4}}=625$ etc.
Now let us consider ${{553}^{3703}}$ . Since 3 is in the unit’s place of 553, let us consider the sequence of unit digits when 3 is raised to natural power.
The power of 3 in unit’s place are
${{3}^{1}}=3$ ${{3}^{9}}=3$
${{3}^{2}}=9$ ${{3}^{10}}=9$
${{3}^{3}}=27=7$ ${{3}^{11}}=7$
${{3}^{4}}=81=1$ ${{3}^{12}}=1$
${{3}^{5}}=243=3$ .
${{3}^{6}}=729=9$ .
${{3}^{7}}=2187=7$ .
${{3}^{8}}=6561=1$ .
Hence, the units place when 3 is raised to any number can be a digit from these four digits which are 1, 3, 7 and 9.
Now, ${{553}^{3703}}$ we need to divide 3703 and find its remainder.
$3703=3700+3$
$=4\times 925+3$
Now 3 is the remainder of 3703. Thus, for ${{3}^{3}}$ the unit place is 7.
$\therefore $ The units place of ${{553}^{3703}}=7$ .
Now let us find the units place of ${{4532}^{828}}$ . We know that the unit place of 4532 is 2. Let us consider the sequence of all the natural powers of 2 and digits at the unit's place.
${{2}^{1}}=2$ ${{2}^{5}}=2$
${{2}^{2}}=4$ ${{2}^{6}}=4$
${{2}^{3}}=8$ ${{2}^{7}}=8$
${{2}^{4}}=6$ ${{2}^{8}}=6$
Now, let us divide 828 and find its remainder
$828=207\times 4+0$
Thus, the units place is ${{2}^{4}}=6$
$\therefore $ The unit place of ${{4532}^{828}}=6$ .
Let us collect the units place of all the digits, multiply them and get their units,
${{125}^{813}}\times {{553}^{3703}}\times {{4532}^{828}}=5\times 7\times 6=210$
Thus, the unit's place of 210 is xero. Hence, we can say that the units place of ${{125}^{813}}\times {{553}^{3703}}\times {{4532}^{828}}$ is zero.
$\therefore $ option (c) is the correct answer.

Note: For shortcut you can check the divisibility of the power by 4.
Remainder when dividing 813 by 4 $=$ 1
Remainder when dividing 3703 by 4 $=$ 3
Remainder when dividing 828 by 4 $=$ 0 i.e. 4
${{125}^{813}}\times {{553}^{3703}}\times {{4532}^{828}}={{125}^{1}}\times {{553}^{3}}\times {{4532}^{4}}$
$={{5}^{1}}\times {{3}^{3}}\times {{2}^{4}}=5\times 7\times 6=210$