Question

$3^{65}$ x $6^{59}$ x $7^{71}$

$\text{find the value in unit place}$

• A. 1
• B. 5
• C. 7
• D. 4

Answer 1

Answer:

D. 4

To find the unit digit in the given product ($3^{65} \times 6^{59} \times 7^{71}$), simply find the units digits of each number separately and multiply them together.

To find the units digit, we need to look at the remainder obtained when the power value is divided by 4.

1. The units digit of $3^{65}$

Powers of 3 repeat in multiples of 4 ($3, 9, 7, 1$).

Divide 65 by 4: $65 \div 4 \rightarrow$ remainder = 1.

Therefore, the units digit of $3^{65}$ is $3^1 = \mathbf{3}$.

2. The unit digit of $6^{59}$

The uniqueness of 6 is that no matter what its power, the unit digit is always 6 ($6^1=6, 6^2=36, 6^3=216 \dots$).

Therefore, the unit digit of $6^{59}$ = $\mathbf{6}$.

3. The units digit of $7^{71}$

The powers of 7 also repeat in multiples of 4 ($7, 9, 3, 1$).

Divide 71 by 4: $71 \div 4 \rightarrow$ remainder = 3.

Therefore, the units digit of $7^{71}$ is the last digit in $7^3$ (i.e. 343), which is 3.

Final step (multiplication):

Now multiply the units digits we have obtained:

$3 \times 6 \times 3 = 54$

The units digit in the number $54$ is 4.