Finding the Unit's Digit of a Mathematical Expression

The question asks for the unit's digit of the number $x$, where $x$ is defined as $x = 23729 \times 78900 + 57492$. To determine the unit's digit of an expression involving addition and multiplication, we can find the unit's digit of each term separately and then combine them based on the operation.

Calculating the Unit's Digit of the Product Term

The first term in the expression is a product: $23729 \times 78900$. The unit's digit of a product is determined by the unit's digits of the numbers being multiplied.

• The unit's digit of the number 23729 is 9.
• The unit's digit of the number 78900 is 0.

The unit's digit of the product $23729 \times 78900$ is the unit's digit of the result of multiplying their unit's digits:

Unit's digit of product = Unit's digit of $(9 \times 0)$

Since $9 \times 0 = 0$, the unit's digit of the product $23729 \times 78900$ is 0.

Calculating the Unit's Digit of the Second Term

The second term in the expression is the number 57492. Its unit's digit is simply the last digit of the number.

• The unit's digit of the number 57492 is 2.

Combining Unit's Digits for the Final Result

The expression for $x$ is the sum of the product term and the second term: $x = (23729 \times 78900) + 57492$. To find the unit's digit of a sum, we look at the unit's digit of the sum of the unit's digits of the terms being added.

The unit's digit of the first term ($23729 \times 78900$) is 0.

The unit's digit of the second term ($57492$) is 2.

The unit's digit of $x$ is the unit's digit of the sum of these unit's digits, which is the unit's digit of $(0 + 2)$.

Based on the calculation, the unit's digit in $x$ is 9.

Comparing with Given Options

Let's compare our result with the provided options:

Our calculated unit's digit is 9, which matches Option 4.

Revision Table: Unit's Digit Rules Summary

Additional Information: Why Unit's Digits Matter

Focusing on the unit's digit is a useful technique for quickly solving certain types of problems involving large numbers, especially in competitive exams. It allows you to avoid lengthy calculations by understanding how the rightmost digit behaves during arithmetic operations.

The unit's digit of a sum or difference depends only on the unit's digits of the numbers being added or subtracted. Similarly, the unit's digit of a product depends only on the unit's digits of the numbers being multiplied.

This property is derived from the base-10 number system, where the value of a digit depends on its position. The unit's digit is in the $10^0$ place. When performing addition or multiplication, the carry-over (in addition) or values from multiplying higher place digits (in multiplication) affect digits to the left of the unit's place but do not change the unit's digit itself, except through the carry-over/product contribution to the unit's place calculation.