HCF and LCM in real life

Knowing when to use HCF and when to use LCM is the real test.

Idea

Use HCF (the biggest "common divisor") when the question asks for the largest size or quantity that divides several given numbers. Typical phrasing:

• "Largest size of square tile fitting a rectangular floor of $24 \times 36$ feet."
• "Greatest number of identical bags possible using $50$ apples and $75$ oranges, with each bag identical."
• "Length of largest measuring rod that exactly measures two given lengths."

Use LCM (the smallest "common multiple") when the question asks for the smallest quantity that fits all given conditions, especially with cycles. Typical phrasing:

• "Smallest number that, when divided by each of $4, 6, 8$, leaves no remainder."
• "Two bells ring every $6$ and $9$ minutes; when do they ring together next?"
• "When will two athletes lapping a track at different times meet at the start?"

A quick test. If the answer must divide both numbers, use HCF. If the answer must contain both numbers (as a multiple), use LCM.

Worked examples

Example 1. A floor is $36$ feet by $48$ feet. The largest square tile that exactly fits?

The tile's side must divide both $36$ and $48$. Largest such number: $\text{HCF}(36, 48) = 12$. So tile size: $12 \times 12$ feet.

Example 2. Two bells ring every $20$ minutes and $30$ minutes. They ring together at $9$ am. Next time?

$\text{LCM}(20, 30) = 60$ minutes. Next together at $10$ am.

Example 3. Largest number of identical gift packs using $24$ chocolates and $36$ candies, each pack identical.

Each pack has $\tfrac{24}{n}$ chocolates and $\tfrac{36}{n}$ candies, both whole numbers. So $n$ divides both , largest is $\text{HCF}(24, 36) = 12$. Each pack: $2$ chocolates, $3$ candies.

Example 4. Smallest number that, when divided by $5, 6, 7$, leaves remainder $0$ each time.

That number is $\text{LCM}(5, 6, 7)$. Since $5, 6, 7$ have no common factor pairwise (except $1$), LCM is $5 \times 6 \times 7 = 210$.

Try it yourself

1. Floor $20 \times 30$ feet. Largest square tile?
2. Two lights blink every $4$ and $6$ seconds, starting together. When together again?
3. Greatest number of identical baskets from $48$ apples and $72$ oranges, each basket identical.
4. Smallest number divisible by $8, 12, 15$.
5. Largest length measuring rod for lengths $40$ cm and $56$ cm.
6. Three friends jog around a track in $6, 8, 12$ minutes per lap. When do all three meet at start?
7. From $90$ pens and $75$ pencils, largest number of identical pouches?
8. Smallest 4-digit number divisible by $24$ and $36$.

Activity

Look at your school timetable for two classes that meet every few days. Use LCM to find the next day both meet.

Measure your bedroom in feet. Find the largest square tile that fits exactly without cutting.