A factor is a number that divides exactly into another number without leaving a remainder.

To find the factors of a number like $90$:

  • Start from $1$ and test incrementally.
  • For each divisor that divides evenly, record both the divisor and the quotient.
  • Stop once the divisor exceeds $\sqrt{90}$.

Step-by-Step: Factors of 90

We start with:

$$ \text{Initial factors: } 1, \ 90 $$

Try 2:

$$ 90 \div 2 = 45 \quad \Rightarrow \text{Add } 2 \text{ and } 45 $$

Try 3:

$$ 90 \div 3 = 30 \quad \Rightarrow \text{Add } 3 \text{ and } 30 $$

Try 4:

$$ 90 \div 4 = 22.5 \quad \text{(Not exact)} \quad \Rightarrow \text{Skip} $$

Try 5:

$$ 90 \div 5 = 18 \quad \Rightarrow \text{Add } 5 \text{ and } 18 $$

Try 6:

$$ 90 \div 6 = 15 \quad \Rightarrow \text{Add } 6 \text{ and } 15 $$

Try 7:

$$ 90 \div 7 \approx 12.857 \quad \text{(Not exact)} \quad \Rightarrow \text{Skip} $$

Try 8:

$$ 90 \div 8 = 11.25 \quad \text{(Not exact)} \quad \Rightarrow \text{Skip} $$

Try 9:

$$ 90 \div 9 = 10 \quad \Rightarrow \text{Add } 9 \text{ and } 10 $$

We can stop here, since: $$ \sqrt{90} \approx 9.49 $$

All Factors of 90

The complete factor list is: $$ 1, \ 2, \ 3, \ 5, \ 6, \ 9, \ 10, \ 15, \ 18, \ 30, \ 45, \ 90 $$

Since $90$ has more than two factors, it is a composite number.

✅ The highest factor (excluding $90$) is: $$ \boxed{45} $$

Summary

  • A factor divides a number exactly with no remainder.
  • A prime number has exactly two factors: $1$ and itself.
  • A composite number has more than two factors.
  • The number $90$ is composite, with factors: $$ 1, \ 2, \ 3, \ 5, \ 6, \ 9, \ 10, \ 15, \ 18, \ 30, \ 45, \ 90 $$