Finding special type of numbers - JavaScript

In the decimal number system, all the real numbers can be divided into two groups:

• Rational Numbers
• Irrational Numbers

For the scope of this problem we will only discuss the rational numbers. All those numbers which can be written in the p/q form (where q ? 0) are called rational numbers, like 14, 4.6, 3.33333... and many more.

The rational numbers can further be divided into two groups:

• Terminating decimal numbers
• Repeating decimal numbers

This categorization is made on the basis of the result obtained upon dividing p by q.

Rule for Terminating Decimals

The rule for this categorization is that:

• We will obtain a terminating decimal number if and only if the prime factors of q are only 2 and 5
• We will obtain a repeating decimal number if the prime factors of q include any other number than 2 or 5

So, we need to write a JavaScript function that takes in a number representing the value q. Our function should return true if for that value we can obtain a terminating decimal number, false otherwise.

Algorithm

The approach is to keep dividing the number by 2 and 5 as long as possible. If we can reduce the number to 1, it means all prime factors were only 2 and 5, so it's a terminating decimal.

Example

const num = 1250;

const isTerminating = num => {
    while(num !== 1) {
        if(num % 2 === 0) {
            num /= 2;
        } else if(num % 5 === 0) {
            num /= 5;
        } else {
            return false;
        }
    }
    return true;
};

console.log(isTerminating(num));

true

How It Works

Let's trace through the example with num = 1250:

• 1250 ÷ 2 = 625
• 625 ÷ 5 = 125
• 125 ÷ 5 = 25
• 25 ÷ 5 = 5
• 5 ÷ 5 = 1

Since we reached 1, all prime factors were only 2 and 5, so 1250 produces a terminating decimal.

Testing with Different Numbers

const isTerminating = num => {
    while(num !== 1) {
        if(num % 2 === 0) {
            num /= 2;
        } else if(num % 5 === 0) {
            num /= 5;
        } else {
            return false;
        }
    }
    return true;
};

// Test cases
console.log("Testing 8:", isTerminating(8));     // true (2^3)
console.log("Testing 10:", isTerminating(10));   // true (2 × 5)
console.log("Testing 6:", isTerminating(6));     // false (2 × 3)
console.log("Testing 15:", isTerminating(15));   // false (3 × 5)

Testing 8: true
Testing 10: true
Testing 6: false
Testing 15: false

Conclusion

A rational number produces a terminating decimal if its denominator has only 2 and 5 as prime factors. Our algorithm efficiently checks this by repeatedly dividing by 2 and 5 until either we reach 1 (terminating) or find another prime factor (non-terminating).