Finding Common Ground

Why This Matters

Sameeksha is building a house. The main room is 12 ft by 16 ft. She wants to cover the floor with square tiles, all the same size. She wants no gaps, no broken tiles, and as few tiles as possible. The side of each tile must be a whole number of feet.

What size tile should she buy?

This is a real puzzle, not just a textbook one. To answer it, she needs a number that fits both 12 and 16 perfectly. A number that divides both with nothing left over. And among all such numbers, she wants the biggest one (the biggest tile means the fewest tiles).

That “biggest number that fits into both” has a name. It is the HCF. It shows up everywhere: tiling floors, packing things into equal groups, cutting ribbons into equal pieces.

There is a partner idea too. When two events repeat on their own cycles — two bells ringing, two friends visiting a shop on different days — when will they happen together? That answer is the LCM.

In this chapter you will learn to find both, by hand and with a quick reliable trick. And, as always on Curriv, you will learn why each method works — not just the steps.

The Big Idea

Two numbers share some factors (numbers that divide both) and some multiples (numbers that both divide into). The Highest Common Factor (HCF) is the biggest factor they share — the biggest number that goes into both. The Lowest Common Multiple (LCM) is the smallest number that both go into. Prime factorisation makes both easy: break each number into primes, then the HCF is the primes they have in common, and the LCM is all the primes (taking the most of each). And there is a beautiful link: HCF × LCM = the product of the two numbers.

Let’s Break It Down

Before we start, let’s quickly refresh three ideas from earlier classes. Everything in this chapter is built on them, so it helps to have them fresh.

Common factors & the HCF

A common factor of two numbers is a number that is a factor of both. Let’s find Sameeksha’s tile size.

For a tile to fit the 12 ft side exactly, its side must be a factor of 12. For it to fit the 16 ft side too, its side must be a factor of 16. So the tile side must be a factor of both 12 and 16 — a common factor.

• Factors of 12: 1, 2, 3, 4, 6, 12
• Factors of 16: 1, 2, 4, 8, 16

Which numbers appear in both lists? They are 1, 2 and 4. These are the common factors. So Sameeksha could use a 1 ft, 2 ft, or 4 ft square tile and have no gaps.

She wants as few tiles as possible, so she picks the biggest one: the 4 ft tile.

That biggest common factor has a name. The Highest Common Factor (HCF) of two or more numbers is the largest of their common factors. (It is also called the Greatest Common Divisor, or GCD — same thing.)

So here, HCF(12, 16) = 4.

A picture makes the “common” part clear. Figure 3.1 below puts the factors of each number in its own circle and the shared ones in the overlap.

Tap to zoom

Why does Sameeksha pick the biggest tile? A bigger tile covers more floor, so fewer tiles are needed. With a 4 ft tile she needs only 12 tiles; with a 1 ft tile she would need 192. The HCF gives the most efficient answer.

Finding HCF (listing and prime factorisation)

We just found the HCF by listing all the factors and spotting the common ones. That works fine for small numbers. But for big numbers, listing every factor is slow, and it is easy to miss one. There is a faster, more reliable way: prime factorisation.

Here is the key idea. When you write a number as a product of primes, every factor of the number is just some “subpart” of those primes. So a common factor must be made of primes that appear in both factorisations.

Let’s see it on 30 and 72.

• 30 = 2 × 3 × 5
• 72 = 2 × 2 × 2 × 3 × 3

Which primes appear in both? A 2 appears in both, and a 3 appears in both. (The number 30 has no second 2, and no extra 3, so we can only take what both share.) Multiply the shared primes:

HCF(30, 72) = 2 × 3 = 6

Figure 3.2 shows this as two circles of prime factors. The HCF lives in the overlap.

Tap to zoom

So the rule for HCF by prime factorisation is short:

HCF = multiply the common primes, taking the minimum number of times each appears.

Why “minimum”? Because the HCF must divide both numbers. If 30 only has one 2, then the HCF can have at most one 2 — otherwise it would not divide 30. So we always take the smaller count.

Let’s break a number into primes with a factor tree and then use it. A factor tree splits a number into two factors, then keeps splitting until only primes are left.

Figure 3.3 shows the factor tree we built for 72 in Step 1.

Tap to zoom

One more useful case: sometimes two numbers share no primes at all. For example, 96 = 2 × 2 × 2 × 2 × 2 × 3 and 275 = 5 × 5 × 11. There is no prime in both lists. Then the only common factor is 1, so HCF = 1. We’ll meet these special numbers again under “co-prime numbers”.

Common multiples & the LCM

Now the partner idea. A common multiple of two numbers is a number that is a multiple of both.

Here is a real example from the textbook. Anshu and Guna make torans (door hangings) from cloth strips placed side by side. Anshu’s strips are 6 cm long, so any toran he makes has a length that is a multiple of 6. Guna’s strips are 8 cm, so his toran length is a multiple of 8. They both want to make torans of the same length. What is the shortest length they can both make?

• Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48, …
• Multiples of 8: 8, 16, 24, 32, 40, 48, 56, …

Numbers in both lists are 24, 48, … These are the common multiples. The smallest one is 24.

That smallest shared multiple has a name. The Lowest Common Multiple (LCM) of two or more numbers is the smallest of their common multiples. So LCM(6, 8) = 24. The shortest matching toran is 24 cm long.

Figure 3.4 shows this on two number lines: the multiples of 6 on top, the multiples of 8 below. The first place a mark lines up on both lines is 24.

Tap to zoom

Notice something: multiples never run out, so the common multiples never run out either (48, 72, 96, … are all common too). There is no highest common multiple. That is why we look for the lowest one.

Finding LCM (listing and prime factorisation)

We just found the LCM by listing multiples until two matched. Again, that is slow for big numbers. And prime factorisation rescues us once more.

The idea flips the HCF rule. A common multiple must contain all the prime building blocks of each number (so that both numbers divide into it). For the lowest one, we take just enough of each prime — the most that appears in either number, but no more.

Let’s do 96 and 360.

• 96 = 2 × 2 × 2 × 2 × 2 × 3 (five 2s, one 3)
• 360 = 2 × 2 × 2 × 3 × 3 × 5 (three 2s, two 3s, one 5)

Now go prime by prime, taking the maximum count of each:

• 2s: 96 has five, 360 has three → take five (so it covers both)
• 3s: 96 has one, 360 has two → take two
• 5s: 96 has none, 360 has one → take one

LCM(96, 360) = 2 × 2 × 2 × 2 × 2 × 3 × 3 × 5 = 1440

Why the maximum? The LCM must be a multiple of 96, so it needs all five of 96’s 2s. It must also be a multiple of 360, so it needs two 3s. Taking the bigger count of each prime covers both numbers at once — and taking no more keeps it as small as possible.

Look back at Figure 3.2: the LCM is simply everything in both circles (the shared primes counted once, plus the leftovers from each side). For 30 and 72 that is 2 × 3 × 5 × 2 × 2 × 3 = 360, so LCM(30, 72) = 360.

So, side by side, the two rules are mirror images:

HCF = common primes, take the minimum of each.

LCM = all primes, take the maximum of each.

The HCF × LCM = product shortcut (and WHY)

Here is a lovely fact that saves a lot of work. For any two numbers:

HCF × LCM = the product of the two numbers

Let’s test it on 12 and 16. We found HCF = 4. Their product is 12 × 16 = 192. So the LCM must be 192 ÷ 4 = 48. Check by listing: multiples of 12 are 12, 24, 36, 48; multiples of 16 are 16, 32, 48 — yes, 48! It works.

This means: if you know the HCF, you can get the LCM instantly with a division, and the other way round. No need to list anything.

Why is this true? Think in primes. Take any prime, say it appears a times in the first number and b times in the second.

• The HCF uses the minimum of a and b.
• The LCM uses the maximum of a and b.

When you multiply HCF × LCM, that prime appears (minimum + maximum) times. But the smaller plus the larger of two numbers is just the same as adding them both: minimum + maximum = a + b. And the product of the two numbers has that prime exactly a + b times too (because you multiply the two factorisations together). Since this matches for every prime, HCF × LCM and the product are made of exactly the same primes — so they are equal.

Careful: this neat rule works for two numbers only. For three or more numbers, HCF × LCM is not equal to the product. Use it just for pairs.

Co-prime numbers

Sometimes two numbers share no common factor except 1. Such a pair is called co-prime (or “relatively prime”).

For example, 8 and 9. Factors of 8 are 1, 2, 4, 8. Factors of 9 are 1, 3, 9. The only number in both lists is 1. So HCF(8, 9) = 1, and 8 and 9 are co-prime.

Note the surprise: co-prime numbers do not have to be prime themselves. 8 and 9 are both composite, yet co-prime, because they share no prime building block (8 is all 2s, 9 is all 3s).

For co-prime numbers there is a neat shortcut. Since HCF = 1, the rule HCF × LCM = product becomes:

For co-prime numbers, LCM = the product of the two numbers (because HCF = 1).

So LCM(8, 9) = 8 × 9 = 72. No listing needed.

Real-life problems

HCF and LCM are tools for real situations. The trick is knowing which one to use. Here is the simple test:

• Are you cutting or grouping things into equal, biggest pieces, with nothing left over? That is HCF (the biggest size that fits into both).
• Are you waiting for two repeating cycles to happen together for the first time? That is LCM (the first shared multiple).

The tiling problem is an HCF problem. Figure 3.5 shows Sameeksha’s room tiled with the 4 ft tiles we found.

Tap to zoom

Now a bell problem, which is an LCM problem.

Here is a comparison table to keep the two ideas straight in your head.

Question	HCF (Highest Common Factor)	LCM (Lowest Common Multiple)
What it is	Biggest number that divides both	Smallest number both divide into
Built from	Factors the numbers share	Multiples the numbers share
Prime rule	Common primes, minimum count	All primes, maximum count
Size	Never bigger than the smaller number	Never smaller than the bigger number
Real-life clue	Cutting / grouping into biggest equal parts	Cycles meeting / happening together

Common Mistakes

Quick Check

Practice Problems

Easy

Medium

Challenge

Summary

• A common factor divides both numbers; the HCF (Highest Common Factor) is the biggest of them — the largest number that fits into both.
• A common multiple is a multiple of both; the LCM (Lowest Common Multiple) is the smallest of them — the smallest number both fit into.
• You can find both by listing (factors for HCF, multiples for LCM), but prime factorisation is faster and more reliable for big numbers.
• HCF = the shared primes, taking the minimum count of each. LCM = all the primes, taking the maximum count of each.
• HCF × LCM = the product of the two numbers — so knowing one lets you find the other by dividing. (This holds for two numbers only.)
• Co-prime numbers share no factor except 1, so their HCF is 1 and their LCM is just their product. They need not be prime themselves (like 8 and 9).
• The HCF is never bigger than the smaller number; the LCM is never smaller than the bigger number.
• Real-life clue: cutting or grouping into biggest equal parts → HCF; repeating cycles meeting together → LCM.

What’s Next

You now have two powerful tools — HCF and LCM — for taming whole numbers. Next, we leave whole numbers behind and zoom between them. In Chapter 12, Another Peek Beyond the Point, we look closely at decimals: the numbers that live in the gaps between whole numbers, the ones you meet every day in money, measurements and marks.