Unit 3 · Topic 03 · Playing with Numbers
Ms. Rao asked Anaya and Kabir to split 12 mangoes and 18 mangoes into equal-sized bags, with no fruit left over, using the biggest bag size possible.
Kabir listed every way to split 12 alone, and every way to split 18 alone, then hunted for a bag size that worked for both.
The biggest number that worked for both piles turned out to have a name: the Highest Common Factor.
"I have 12 mangoes and 18 guavas," said Ms. Rao. "I want to pack them into identical gift bags — same number of bags, no fruit left over, and I want the FEWEST bags possible." Kabir wrote out every number that divides 12 evenly: 1, 2, 3, 4, 6, 12. Then every number that divides 18 evenly: 1, 2, 3, 6, 9, 18.
"Those are the FACTORS of each number," said Ms. Rao. "Now, which numbers appear in BOTH lists?" Anaya circled them: 1, 2, 3, and 6 appeared in both. "Those are the common factors — sizes that work for splitting either pile evenly."
"To use the fewest bags, I want the LARGEST common factor," Ms. Rao continued. Kabir looked at the circled numbers — 1, 2, 3, 6 — and picked the biggest: 6. "So 6 mangoes per bag and 6 guavas per bag. That gives 2 bags of mangoes and 3 bags of guavas, and every bag has exactly 6 fruits."
"That biggest common factor, 6, is called the Highest Common Factor, or HCF, of 12 and 18," said Ms. Rao. "It's the largest number that divides two or more numbers exactly, with nothing left over." Anaya tried a harder pair: 20 and 30. Factors of 20: 1,2,4,5,10,20. Factors of 30: 1,2,3,5,6,10,15,30. Common factors: 1,2,5,10. The HCF was 10 — the biggest number both piles could be evenly split by.
A factor of a number divides it exactly, leaving no remainder. Every number has 1 and itself as factors.
A common factor of two or more numbers is a number that divides all of them exactly.
The Highest Common Factor (HCF) is the largest of these common factors — the biggest number that divides every given number exactly.
To find the HCF by listing: write out all factors of each number, circle the ones shared by every list, and pick the largest circled number.
List all the factors of 16 and of 24. Circle the common factors and identify the HCF.
Explain in your own words why every number's HCF with itself is the number itself.
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