Problem
The keyword “what is the factor of 7” is interpreted as: What are the (positive) integer factors of 7?
Definition
Factor (divisor): An integer \(d\) is a factor of an integer \(n\) if there exists an integer \(k\) such that \( n = d \cdot k \). Equivalently, dividing \(n\) by \(d\) gives remainder 0.
Finding the factors of 7
1) Use factor pairs
A factor pair of 7 is a pair of integers whose product is 7. Since 7 is positive, positive factor pairs must multiply to 7:
\[ 7 = 1 \cdot 7 \]
This produces the positive factors \(1\) and \(7\).
2) Verify by quick divisibility checks up to \( \sqrt{7} \)
To find all positive factors, it is enough to test divisors from 1 up to \( \sqrt{7} \), because factors come in pairs \(d\) and \( \frac{7}{d} \). Since
\[ \sqrt{7} \approx 2.645\ldots \]
only \(d=1\) and \(d=2\) need to be tested.
| Test divisor \(d\) | Does \(d\) divide 7? | Reason |
|---|---|---|
| 1 | Yes | \(7 = 1 \cdot 7\) (remainder 0) |
| 2 | No | \(7\) is odd, so it is not divisible by \(2\) |
Therefore, the only positive factors of 7 are \(1\) and \(7\).
What this says about 7
A positive integer greater than 1 is prime if its only positive factors are 1 and itself. Since 7 has exactly the positive factors \(1\) and \(7\), 7 is a prime number. Its prime factorization is simply
\[ 7 = 7 \]
Extension: If negative factors are included, the integer factors of 7 are \( \pm 1 \) and \( \pm 7 \), because \(7 = (-1)\cdot(-7)\) as well as \(7 = 1\cdot 7\).
Visualization: the only positive rectangle for 7 is \(1 \times 7\)
Conclusion
The positive factors of 7 are \(1\) and \(7\); equivalently, 7 is divisible only by 1 and itself, so 7 is prime.