Finding the Unit Digit of the Product

The problem asks for the unit digit of the product obtained by multiplying three terms: $\{(341)^{491} \times (625)^{317} \times (6374)^{1793}\}$. To find the unit digit of the entire product, we first need to determine the unit digit of each individual term.

Unit Digit Analysis of Powers

The unit digit of a number raised to a power depends on the unit digit of the base number and the exponent. We can observe patterns, known as the cyclicity of unit digits.

Unit Digit of $(341)^{491}$

The base number is 341, which has a unit digit of 1.

Rule: Any number ending in 1, when raised to any positive integer power, will always have a unit digit of 1.

Example: $1^1=1$, $11^2=121$, $21^3=9261$.

Therefore, the unit digit of $(341)^{491}$ is 1.

Unit Digit of $(625)^{317}$

The base number is 625, which has a unit digit of 5.

Rule: Any number ending in 5, when raised to any positive integer power, will always have a unit digit of 5.

Example: $5^1=5$, $5^2=25$, $15^3=3375$.

Therefore, the unit digit of $(625)^{317}$ is 5.

Unit Digit of $(6374)^{1793}$

The base number is 6374, which has a unit digit of 4.

Let's look at the pattern of the unit digits of powers of 4:

• $(...4)^1 \rightarrow 4$
• $(...4)^2 \rightarrow 6$
• $(...4)^3 \rightarrow 4$
• $(...4)^4 \rightarrow 6$

The pattern of the unit digits is (4, 6), which repeats every 2 powers.

Rule: If the exponent is odd, the unit digit is 4. If the exponent is even, the unit digit is 6.

The exponent here is 1793.

Since 1793 is an odd number, the unit digit of $(6374)^{1793}$ is 4.

Product Unit Digit Calculation

Now, to find the unit digit of the product $\{(341)^{491} \times (625)^{317} \times (6374)^{1793}\}$, we multiply the unit digits of each term:

Unit digit of the product = (Unit digit of $(341)^{491}$) $\times$ (Unit digit of $(625)^{317}$) $\times$ (Unit digit of $(6374)^{1793}$)

Unit digit of the product = $1 \times 5 \times 4$

Unit digit of the product = $20$

The unit digit of 20 is 0.

Final Unit Digit

Therefore, the unit digit in the product $\{(341)^{491} \times (625)^{317} \times (6374)^{1793}\}$ is 0.