Exact arithmetic and estimation · Foundation
הוצג| English | משמעות |
|---|---|
| prime factor/praɪm ˈfæktə/ | גורם ראשוני |
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?
רמת קושי 1/5 · 100 מטבעות + בונוס קטן בהשלמה הראשונה היום
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.
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.
רמת קושי 2/5 · 140 מטבעות + בונוס קטן בהשלמה הראשונה היום
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.
רמת קושי 2/5 · 140 מטבעות + בונוס קטן בהשלמה הראשונה היום
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.
רמת קושי 1/5 · 100 מטבעות + בונוס קטן בהשלמה הראשונה היום
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.
רמת קושי 2/5 · 140 מטבעות + בונוס קטן בהשלמה הראשונה היום
Use denominator 12: 3/4+5/6=9/12+10/12=19/12.
Match each quantity to how prime factorisation produces it.
רמת קושי 2/5 · 140 מטבעות + בונוס קטן בהשלמה הראשונה היום
The HCF keeps only shared prime factors at their lower powers; the LCM keeps every prime at its higher power.
In your course
- Pearson Edexcel International GCSE Mathematics A (4MA1), Foundation tier, specification section 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.
Write from an HCF of 26
- 52 and 78 are the two totals.
- Find the HCF and the LCM.
Find the HCF and the LCM of 52 and 78.
26 times 156 equals 52 times 78. The shared prime power is 2 times 13.
What is the HCF of 52 and 78?
רמת קושי 1/5 · 100 מטבעות + בונוס קטן בהשלמה הראשונה היום
The shared factors give 26. The unshared 3 stays in the LCM. Other options — 18 (the lesson example's HCF of 72 and 90); 78 (the larger total).