Enter a positive integer to list its factors and factor pairs, see the prime factorization, and walk a factor tree. For a prime, the result simply says so and shows the pair (1, n).
Factor-pairs rectangle activity
This original check-and-explain activity turns each factor pair into the side lengths of a whole-number rectangle. It is designed for learners to reason on paper or grid paper first and use the calculator afterward to check whether a pair was missed.
- For each number below, predict whether it is prime, composite, or a perfect square.
- Draw or list every different rectangle with whole-number side lengths and the stated area. Treat a 3-by-16 rectangle and a 16-by-3 rectangle as the same rectangle.
- Explain how you know the list is complete. A useful checkpoint is √n.
- Enter the number in the calculator and compare its factor-pair and prime factorization results with your work.
| Area | Factor-pair answer key | Classification | Prime factorization |
|---|---|---|---|
| 24 | 1 × 24, 2 × 12, 3 × 8, 4 × 6 | Composite | 2 × 2 × 2 × 3 |
| 32 | 1 × 32, 2 × 16, 4 × 8 | Composite | 2 × 2 × 2 × 2 × 2 |
| 36 | 1 × 36, 2 × 18, 3 × 12, 4 × 9, 6 × 6 | Composite; perfect square | 2 × 2 × 3 × 3 |
| 37 | 1 × 37 | Prime | 37 |
| 48 | 1 × 48, 2 × 24, 3 × 16, 4 × 12, 6 × 8 | Composite | 2 × 2 × 2 × 2 × 3 |
The Common Core 4.OA.B.4 standard includes finding all factor pairs for whole numbers from 1 through 100 and deciding whether a number is prime or composite. Illustrative Mathematics' factor-pairs materials also connect factor pairs with rectangles built from whole-number side lengths. These references support the mathematical context; this worksheet and answer key are CalculatorBuddy's own examples and have not been classroom tested.
Teacher or study prompts
- Which number has the most different rectangles? Does that necessarily mean it has the largest value?
- Why does 36 include 6 × 6 only once?
- Which result proves immediately that 37 is prime?
- Compare the factor pairs with the prime factorization. What different question does each representation answer?
- Extension: choose a number from 1 to 100 that has exactly three different factor pairs, then verify it with the calculator.
Examples
Default 120
Factors are 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120. Pairs run (1, 120) through (10, 12). Prime factorization is 2 × 2 × 2 × 3 × 5.
Page example 20
Factors 1, 2, 4, 5, 10, 20, pairs (1, 20) (2, 10) (4, 5), and 20 = 2 × 2 × 5.
Prime 7
7 is a prime number. Its only factor pair is (1, 7).
Formula
An integer a is a factor of n when n ÷ a has remainder 0. The positive factors come in pairs (a, n/a) found by testing a = 1, 2, … up to √n.
Prime factorization. Divide out the smallest prime repeatedly:
120 = 2 × 2 × 2 × 3 × 5
Factor tree (same splits, quotient on the left, prime on the right):
120
| \
60 2
| \
30 2
| \
15 2
| \
5 3
1 is neither prime nor composite; its only pair is (1, 1). A prime has exactly two positive factors, 1 and itself.
For a systematic search, stop after the divisor passes √n. For example, √48 is about 6.93, so testing 1 through 6 finds every pair: 1 × 48, 2 × 24, 3 × 16, 4 × 12, and 6 × 8. The next whole number, 7, is already beyond √48; continuing would only find the same pairs in reverse.
Simplifying a fraction a/b uses GCF(a, b), not the full factor list. Open the GCF calculator when you want the greatest common factor of several numbers, or the LCM calculator for the least common multiple.
One number, or several numbers?
This calculator lists factors of one integer. To find a divisor shared by several numbers, use the GCF Calculator. To find the first positive multiple they share, use the LCM Calculator. The distinction is useful when moving from factor pairs to grouping or repeating-schedule problems.