- A factor pair of \(N\) is a pair \((a, b)\) such that \(a \times b = N\) and \(a \le b\).
- Systematic identification involves finding divisors \(a\) up to \(\sqrt{N}\) and calculating \(b = N/a\).
- The factor pairs are:
- 24: \((1, 24), (2, 12), (3, 8), (4, 6)\)
- 36: \((1, 36), (2, 18), (3, 12), (4, 9), (6, 6)\)
- 40: \((1, 40), (2, 20), (4, 10), (5, 8)\)
- 63: \((1, 63), (3, 21), (7, 9)\)
- 72: \((1, 72), (2, 36), (3, 24), (4, 18), (6, 12), (8, 9)\)
Explanation
- Methodology for Finding Factor Pairs
A factor pair of a positive integer \(N\) is a pair of positive integers \((a, b)\) such that \(a \times b = N\). By convention, we list these pairs such that \(a \le b\) to ensure each unique pair is counted once. To find all factor pairs, we test integers \(a\) starting from \(1\) up to \(\sqrt{N}\). For every \(a\) that divides \(N\) without a remainder, the corresponding value \(b = N/a\) forms the pair \((a, b)\).
- Factor Pairs for 24
For \(N = 24\), we test divisors \(a\) from \(1\) to \(\lfloor\sqrt{24}\rfloor = 4\):
- \(a = 1 \implies b = 24/1 = 24\). Pair: \((1, 24)\)
- \(a = 2 \implies b = 24/2 = 12\). Pair: \((2, 12)\)
- \(a = 3 \implies b = 24/3 = 8\). Pair: \((3, 8)\)
- \(a = 4 \implies b = 24/4 = 6\). Pair: \((4, 6)\)
The factor pairs are \((1, 24), (2, 12), (3, 8), (4, 6)\).
- Factor Pairs for 36
For \(N = 36\), we test divisors \(a\) from \(1\) to \(\lfloor\sqrt{36}\rfloor = 6\):
- \(a = 1 \implies b = 36/1 = 36\). Pair: \((1, 36)\)
- \(a = 2 \implies b = 36/2 = 18\). Pair: \((2, 18)\)
- \(a = 3 \implies b = 36/3 = 12\). Pair: \((3, 12)\)
- \(a = 4 \implies b = 36/4 = 9\). Pair: \((4, 9)\)
- \(a = 5\) (does not divide 36)
- \(a = 6 \implies b = 36/6 = 6\). Pair: \((6, 6)\)
The factor pairs are \((1, 36), (2, 18), (3, 12), (4, 9), (6, 6)\).
- Factor Pairs for 40
For \(N = 40\), we test divisors \(a\) from \(1\) to \(\lfloor\sqrt{40}\rfloor = 6\):
- \(a = 1 \implies b = 40/1 = 40\). Pair: \((1, 40)\)
- \(a = 2 \implies b = 40/2 = 20\). Pair: \((2, 20)\)
- \(a = 3\) (does not divide 40)
- \(a = 4 \implies b = 40/4 = 10\). Pair: \((4, 10)\)
- \(a = 5 \implies b = 40/5 = 8\). Pair: \((5, 8)\)
- \(a = 6\) (does not divide 40)
The factor pairs are \((1, 40), (2, 20), (4, 10), (5, 8)\).
- Factor Pairs for 63
For \(N = 63\), we test divisors \(a\) from \(1\) to \(\lfloor\sqrt{63}\rfloor = 7\):
- \(a = 1 \implies b = 63/1 = 63\). Pair: \((1, 63)\)
- \(a = 2\) (does not divide 63)
- \(a = 3 \implies b = 63/3 = 21\). Pair: \((3, 21)\)
- \(a = 4, 5, 6\) (do not divide 63)
- \(a = 7 \implies b = 63/7 = 9\). Pair: \((7, 9)\)
The factor pairs are \((1, 63), (3, 21), (7, 9)\).
- Factor Pairs for 72
For \(N = 72\), we test divisors \(a\) from \(1\) to \(\lfloor\sqrt{72}\rfloor = 8\):
- \(a = 1 \implies b = 72/1 = 72\). Pair: \((1, 72)\)
- \(a = 2 \implies b = 72/2 = 36\). Pair: \((2, 36)\)
- \(a = 3 \implies b = 72/3 = 24\). Pair: \((3, 24)\)
- \(a = 4 \implies b = 72/4 = 18\). Pair: \((4, 18)\)
- \(a = 5\) (does not divide 72)
- \(a = 6 \implies b = 72/6 = 12\). Pair: \((6, 12)\)
- \(a = 7\) (does not divide 72)
- \(a = 8 \implies b = 72/8 = 9\). Pair: \((8, 9)\)
The factor pairs are \((1, 72), (2, 36), (3, 24), (4, 18), (6, 12), (8, 9)\).
- Final Summary
The requested factor pairs are:
A\(24: (1, 24), (2, 12), (3, 8), (4, 6)\)
B\(36: (1, 36), (2, 18), (3, 12), (4, 9), (6, 6)\)
C\(40: (1, 40), (2, 20), (4, 10), (5, 8)\)
D\(63: (1, 63), (3, 21), (7, 9)\)
E\(72: (1, 72), (2, 36), (3, 24), (4, 18), (6, 12), (8, 9)\)
The final answer is complete.
Examples
Factor pairs are essential in real-life logistics and design. For example, if you are designing a rectangular garden with a fixed area of 72 square meters, knowing the factor pairs allows you to determine the possible integer dimensions for your plot (e.g., \(1 \times 72, 2 \times 36, 3 \times 24, 4 \times 18, 6 \times 12, 8 \times 9\)). This helps in planning irrigation layouts, fencing, and crop arrangement to best utilize the available space efficiently.