What Is a Prime Number?

A prime number is a number that has exactly two distinct positive factors:

  • (1)
  • The number itself

If a number has more than two factors, it is considered a composite number.

Is 61 Prime? Step-by-Step

Let’s test whether any number from 2 up to ( \sqrt{61} \approx 7.81 ) divides 61 evenly.

We stop testing once we exceed the square root.

Test Small Divisors:

  • (61 \div 2 = 30.5) → ❌ not divisible
  • (61 \div 3 = 20.33) → ❌ not divisible
  • (61 \div 4 = 15.25) → ❌ not divisible
  • (61 \div 5 = 12.2) → ❌ not divisible
  • (61 \div 6 \approx 10.17) → ❌ not divisible
  • (61 \div 7 \approx 8.71) → ❌ not divisible

No whole numbers divide 61 other than 1 and itself.

Conclusion

Since 61 has exactly two factors — (1) and (61) — it is a prime number.

$$ \boxed{61\ \text{is a prime number.}} $$

Summary

  • Prime numbers have exactly two factors: (1) and the number itself.
  • We tested every number up to ( \sqrt{61} ) and found no other factors.
  • Therefore, 61 is a prime number.