Exact arithmetic and estimation · Higher
導入済み| English | 意味 |
|---|---|
| prime factor/praɪm ˈfæktə/ | 素因数 |
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.
The method
- 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.
Which statement correctly relates HCF and LCM of two positive integers?
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
The highest common factor must divide each number, and the lowest common multiple must be a multiple of each. Only when the numbers share no factor does the LCM equal the product.
A worked case
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.
難易度 2/5 · 140 コイン + 今日初回完了時に小ボーナス付与
48=2⁴×3 and 60=2²×3×5. The shared smallest powers give HCF=2²×3=12.
Where it goes wrong
- The HCF divides both numbers; the LCM is a multiple of both.
- They answer different questions.
- A decimal estimate is not an exact fraction.
Not true: The HCF of two positive integers is larger than either integer. The HCF divides both numbers, so it can never be larger than the smaller one: the HCF of 72 and 90 is 18.
Find the LCM of 8 and 12.
難易度 2/5 · 140 コイン + 今日初回完了時に小ボーナス付与
8=2³ and 12=2²×3. The largest prime powers give LCM=2³×3=24.
The HCF of two positive integers is larger than either integer.
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
The HCF divides both numbers, so it can never be larger than the smaller one: the HCF of 72 and 90 is 18.
Applying it
- For a non-calculator paper, keep fractions exact and show cancellation.
- For a calculator paper, enter the full expression and compare with your estimate.
Work out 3/4 + 5/6 as a decimal. Give it to 3 significant figures.
難易度 2/5 · 140 コイン + 今日初回完了時に小ボーナス付与
Use denominator 12: 3/4+5/6=9/12+10/12=19/12.
Match each quantity to how prime factorisation produces it.
難易度 2/5 · 140 コイン + 今日初回完了時に小ボーナス付与
The HCF keeps only shared prime factors at their lower powers; the LCM keeps every prime at its higher power.
In your course
- AQA GCSE Mathematics (8300), Higher tier, specification section 3.1.
- Prime factor: A prime number that divides the integer exactly.
- Prime factors reveal shared structure.
- Watch for: the HCF divides both numbers; the LCM is a multiple of both.