Solution - Fraction to decimal (repeating)

$$\frac{65780}{2598960}$$

At each cycle: multiply remainder by 10, divide by denominator, record digit and new remainder.

$$\frac{65780}{2598960}$$

This cycle contributes one new decimal digit.

$$657800$$

This cycle contributes one new decimal digit.

$$1380080$$

This cycle contributes one new decimal digit.

$$806000$$

This cycle contributes one new decimal digit.

$$263120$$

This cycle contributes one new decimal digit.

$$32240$$

This cycle contributes one new decimal digit.

$$322400$$

This cycle contributes one new decimal digit.

$$625040$$

This cycle contributes one new decimal digit.

$$1052480$$

This cycle contributes one new decimal digit.

$$128960$$

This cycle contributes one new decimal digit.

$$1289600$$

This cycle contributes one new decimal digit.

$$2500160$$

This cycle contributes one new decimal digit.

$$1610960$$

This cycle contributes one new decimal digit.

$$515840$$

This cycle contributes one new decimal digit.

$$2559440$$

Cycle transcript shortened for readability; continue using the same pattern.

$$14$$

A remainder repeated, so digits from that point form the repeating cycle.

$$0.02\left(531012404961984793917567026810724289715886354541816726690676270508203281312525010004001600640256102440976390556222488995598239295718287314925970388155262104841936774709883953581432573029211684673869547819127651060424169667867146858743497398959583833533413365346138455382152861144457783113245298119247699079631852741096438575430172068827\right)$$

A repeated remainder appears, so the repeating block is $$531012404961984793917567026810724289715886354541816726690676270508203281312525010004001600640256102440976390556222488995598239295718287314925970388155262104841936774709883953581432573029211684673869547819127651060424169667867146858743497398959583833533413365346138455382152861144457783113245298119247699079631852741096438575430172068827$$.

$$0.02\left(531012404961984793917567026810724289715886354541816726690676270508203281312525010004001600640256102440976390556222488995598239295718287314925970388155262104841936774709883953581432573029211684673869547819127651060424169667867146858743497398959583833533413365346138455382152861144457783113245298119247699079631852741096438575430172068827\right)$$

Conclusion: the fraction converts to a repeating decimal.

$$0.02\left(531012404961984793917567026810724289715886354541816726690676270508203281312525010004001600640256102440976390556222488995598239295718287314925970388155262104841936774709883953581432573029211684673869547819127651060424169667867146858743497398959583833533413365346138455382152861144457783113245298119247699079631852741096438575430172068827\right)$$

Study tip: keep parentheses around repeating digits to preserve exact value.

$$0.02\left(531012404961984793917567026810724289715886354541816726690676270508203281312525010004001600640256102440976390556222488995598239295718287314925970388155262104841936774709883953581432573029211684673869547819127651060424169667867146858743497398959583833533413365346138455382152861144457783113245298119247699079631852741096438575430172068827\right)$$

Write the repeating decimal with parentheses: $$0.02\left(531012404961984793917567026810724289715886354541816726690676270508203281312525010004001600640256102440976390556222488995598239295718287314925970388155262104841936774709883953581432573029211684673869547819127651060424169667867146858743497398959583833533413365346138455382152861144457783113245298119247699079631852741096438575430172068827\right)$$.