Question

What is the unit digit in the product of $\left( {{3}^{65}}\times {{6}^{59}}\times {{7}^{71}} \right)$ ?

Answer 1

To find the unit digit in the product of $\left( {{3}^{65}}\times {{6}^{59}}\times {{7}^{71}} \right)$ , we must consider the unit digit of each term. Let us first consider ${{3}^{65}}$ . Using the formulas ${{\left( {{a}^{m}} \right)}^{n}}={{a}^{mn}}$ and ${{a}^{m}}\times {{a}^{n}}={{a}^{m+n}}$ we can write ${{3}^{65}}={{\left( {{3}^{4}} \right)}^{16}}\times 3$ . Now, we must find the unit digit of this. Next, let us consider ${{6}^{59}}$ . We know that 6 to the power of any number has unit digit 6. Hence, we will get its unit digit as 6. Lastly, consider ${{7}^{71}}$ . We can write this as ${{7}^{71}}={{\left( {{7}^{4}} \right)}^{17}}\times {{7}^{3}}$ . Now, multiply the unit digits of all the three terms and find the unit digit of the resulting number.

Complete step-by-step answer:
We need to find the unit digit in the product of $\left( {{3}^{65}}\times {{6}^{59}}\times {{7}^{71}} \right)$ .
Let us first consider ${{3}^{65}}$
We know that ${{\left( {{a}^{m}} \right)}^{n}}={{a}^{mn}}$ and ${{a}^{m}}\times {{a}^{n}}={{a}^{m+n}}$
$\Rightarrow {{3}^{65}}={{\left( {{3}^{4}} \right)}^{16}}\times 3$
Now, let us see the unit digit of the above. That is
Unit digit of $\left[ {{\left( {{3}^{4}} \right)}^{16}}\times 3 \right]$
Let us expand ${{3}^{4}}$ . We will get
Unit digit of $\left[ {{\left( 3\times 3\times 3\times 3 \right)}^{16}}\times 3 \right]$
This be written as
Unit digit of $\left[ {{\left( 9\times 9 \right)}^{16}}\times 3 \right]$
We know that $9\times 9=81$ . Hence, the above form becoms
Unit digit of $\left[ {{\left( 81 \right)}^{16}}\times 3 \right]$
This is same as
$\text{Unit digit of}{{\left( 81 \right)}^{16}}\times \text{Unit digit of }3$
We know that 1 to the power of any number is always 1. Hence,
$\begin{aligned} & \text{Unit digit of}{{\left( 81 \right)}^{16}}\times \text{Unit digit of }3 \\\ & =1\times 3 \\\ & =3...(i) \\\ \end{aligned}$
Now, let us consider ${{6}^{59}}$ .
We know that 6 to the power of any number has unit digit 6. For example
$\begin{aligned} & {{6}^{1}}=6 \\\ & {{6}^{2}}=36 \\\ & {{6}^{3}}=216 \\\ & {{6}^{4}}=1296 \\\ & {{6}^{5}}=7776 \\\ & ... \\\ \end{aligned}$
Hence, unit digit of ${{6}^{59}}=6...(ii)$
Let us now consider ${{7}^{71}}$ .
We know that ${{\left( {{a}^{m}} \right)}^{n}}={{a}^{mn}}$ and ${{a}^{m}}\times {{a}^{n}}={{a}^{m+n}}$
$\Rightarrow {{7}^{71}}={{\left( {{7}^{4}} \right)}^{17}}\times {{7}^{3}}$
Now, let us observe the unit digit.
$\text{Unit digit of }{{7}^{71}}=\text{Unit digit of }\left[ {{\left( {{7}^{4}} \right)}^{17}}\times {{7}^{3}} \right]$
When we expand ${{7}^{4}}$ , we will get
$\text{Unit digit of }{{7}^{71}}=\text{Unit digit of }\left[ {{\left( 7\times 7\times 7\times 7 \right)}^{17}}\times {{7}^{3}} \right]$
We can write this as
$\text{Unit digit of }{{7}^{71}}=\text{Unit digit of }\left[ {{\left( 49\times 49 \right)}^{17}}\times \left( 7\times 7\times 7 \right) \right]$
$\Rightarrow \text{Unit digit of }{{7}^{71}}=\text{Unit digit of }\left[ {{\left( 2401 \right)}^{17}}\times 343 \right]$
We can expand the above form as
$\text{Unit digit of }{{7}^{71}}=\text{Unit digit of }{{\left( 2401 \right)}^{17}}\times \text{Unit digit of 343}$
We know that 1 to the power of any number is always 1. Hence,
$\text{Unit digit of }{{7}^{71}}=1\times 3=3...(iii)$
Now, let us multiply (i),(ii) and (iii). We will get

& \text{Unit digit of}\left( {{3}^{65}}\times {{6}^{59}}\times {{7}^{71}} \right)=\text{Unit digit of }\left( 3\times 6\times 3 \right) \\\ & =\text{Unit digit of }\left( 54 \right) \\\ & =4 \\\ \end{aligned}$$ Hence, the unit digit in the product of $\left( {{3}^{65}}\times {{6}^{59}}\times {{7}^{71}} \right)$ is 4. **Note:** One must know what a unit digit is when solving this question. Unit digit is the number in one’s place in the place value, that is, the last digit of a number. You must know the rules of exponents to split the numbers. When large exponents are given, you must split it and solve for each part. Just like the last digit of any power of 6 is 6, the last digit of any power of 5 is also 5 and for 1 also it is 1 itself. All other numbers, i.e 2, 3, 4, 7, 8 and 9 have certain numbers repeating themselves. For example, the last digit of powers 1, 2, 3, 4 of 9 repeat in the cycle as 1, 9, 1, 9. Knowing these is a great help in the exams.