Q.Program 6-14: Program to check if the input number is prime or not.

Finding Factors: What Does It Really Mean?

Imagine you have 12 identical marbles. You want to arrange them in a neat rectangle — rows and columns, no gaps, no marbles left over. How many different rectangles can you make?

You could do 1 row of 12 marbles. That's $1 \times 12$.

You could do 2 rows of 6 marbles. That's $2 \times 6$.

You could do 3 rows of 4 marbles. That's $3 \times 4$.

You could also do 4 rows of 3, 6 rows of 2, or 12 rows of 1 — but those are just the same rectangles turned sideways.

The numbers that multiply together to give 12 — 1, 2, 3, 4, 6, and 12 — are called the factors of 12. A factor is simply a whole number that divides another number exactly, leaving no remainder.

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The Precise Definition

A factor of a given whole number $n$ is any whole number $f$ such that $n \div f$ is also a whole number. Equivalently, there exists some whole number $k$ such that $f \times k = n$.

For example, 3 is a factor of 12 because $12 \div 3 = 4$ (a whole number), and indeed $3 \times 4 = 12$.

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How to Find All Factors of a Number

The most reliable method is pairwise checking — test every whole number from 1 up to the square root of $n$. For each number that divides $n$ evenly, you get a pair: the divisor and the quotient.

Let's find all factors of 24.

• Start at 1: $24 \div 1 = 24$ → pair (1, 24)
• 2: $24 \div 2 = 12$ → pair (2, 12)
• 3: $24 \div 3 = 8$ → pair (3, 8)
• 4: $24 \div 4 = 6$ → pair (4, 6)
• 5: $24 \div 5 = 4.8$ — not a whole number, so 5 is not a factor
• 6: we already have it from the pair with 4
• Stop at $\sqrt{24} \approx 4.9$ — we've already checked up to 4, so we're done.

The factors of 24 are: 1, 2, 3, 4, 6, 8, 12, 24.

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Why This Matters …