Exact arithmetic and estimation · Higher
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| prime factor/praɪm ˈfæktə/ | 质因数 | zhì yīn shù |
Can we pack without leftovers?
- A supplier packs 72 pencils and 90 pens into identical gift bags. How can we avoid leftovers?
- This lesson studies prime factor 质因数: A prime number that divides the integer exactly.
Choose the mathematical structure
- Prime factors reveal shared structure. Use the smallest common prime powers for the HCF and the largest for the LCM. Estimate before calculating; use brackets to preserve the order of operations.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines prime factor?
A prime number that divides the integer exactly.
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
72=2^3×3^2 and 90=2×3^2×5. Their HCF is 2×9=18. Make 18 bags with 4 pencils and 5 pens each. Their LCM is 2^3×3^2×5=360. A prime integer is greater than 1 and has exactly two positive factors. The number 1 is not prime; a composite positive integer greater than 1 has more than two positive factors. Every integer greater than 1 has a unique product of prime factors apart from their order. For example, 60=2²×3×5. A factor divides a number; a multiple is the result of multiplying it by an integer. HCF uses common smallest powers, while LCM uses largest powers.
Exact arithmetic and estimation
Prime factors reveal shared structure
Combine the common powers 2 and 9 to justify 18 identical bags.
Find the HCF of 48 and 60.
48=2⁴×3 and 60=2²×3×5. The shared smallest powers give HCF=2²×3=12.
Test a tempting shortcut
- The HCF divides both numbers; the LCM is a multiple of both. They answer different questions. A decimal estimate is not an exact fraction.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
The HCF of two positive integers is always larger than either integer. This claim is false. Explain which definition or assumption it violates.
Find the LCM of 8 and 12.
8=2³ and 12=2²×3. The largest prime powers give LCM=2³×3=24.
The HCF of two positive integers is always larger than either integer.
The HCF divides both numbers; the LCM is a multiple of both. They answer different questions. A decimal estimate is not an exact fraction.
Interpret a new situation
- For a non-calculator paper, keep fractions exact and show cancellation. For a calculator paper, enter the full expression and compare with your estimate.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Work out 3/4 + 5/6 as a decimal.
Use denominator 12: 3/4+5/6=9/12+10/12=19/12.
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the question.
Use this in your course
- 8300 · Higher · 3.1. Match the target tier and specification before assigning extensions.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
A prime number that divides the integer exactly. Choose the relationship, show the method, check its assumptions and interpret the result.