Finding the 80th Digit to the Right of the Decimal Point in 7/27

When dealing with the decimal representation of 7/27, one might ask how to find the 80th digit to the right of the decimal point. This problem involves understanding the repeating cycle and using modulo operation. This article will guide you through the process step by step.

Calculating the Decimal Representation of 7/27

First, let's determine the decimal representation of 7/27 through long division:

Perform the division 7 ÷ 27:

7 ÷ 27 0.259259259…

Identify the repeating cycle:

The repeating part is 259.

Understanding the Repeating Cycle

The decimal representation of 7/27 is a repeating decimal with a repeating block of 259. The order of 10 mod 27 is 3, meaning that the decimal repeats every 3 digits. This means the digits 259 form the repeating block.

Using Modulo Operation to Determine the 80th Digit

To find the 80th digit to the right of the decimal point, we need to determine the position of 80 within the repeating cycle. This can be done using the modulo operation:

Calculate 80 mod 3:

80 ÷ 3 26 remainder 2

Find the corresponding digit:

Since the remainder is 2, the 80th digit is the second digit in the repeating block, which is 5.

Therefore, the 80th digit to the right of the decimal point is 5.

Alternative Approach Using Long Division

An alternative approach involves long division to verify the repeating decimal:

Start with 7/27:

7 0 × 27 7 70 2 × 27 16 160 5 × 27 25 250 9 × 27 7 70 2 × 27 16 (repeats)

Thus, the decimal is 0.259259....

The repeating block is 259.

Since 100 33 × 3 1, the 100th digit is the first digit of the repeating block, which is 2.

The 99th digit is 9, and the 100th digit is 2.

The 80th digit is the same as the 2nd digit in the repeating block, which is 5.

Conclusion

The 80th digit to the right of the decimal point in the decimal representation of 7/27 is 5.