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NCERT Solutions Ganita Prakash Chapter 5 In-text Questions — Prime Factorisation

Book page 117 to 120 Updated on2026-09-05

Q1.
Teacher: Are 56 and 63 co-prime? Anshu: I can write 56 = 14 × 4 and 63 = 21 × 3 … there are no common factors. The numbers are co-prime. Guna: Hold on. I can also write 56 = 7 × 8 and 63 = 9 × 7 … so they are not co-prime. Clearly Guna is right, as 7 is a common factor. But where did Anshu go wrong?
Answer

Anshu's mistake was to think that one way of splitting a number shows all its factors. It does not.

56 = 14 × 4 → tells us 14 and 4 are factors of 56
But the full list is: 1, 2, 4, 7, 8, 14, 28, 56

63 = 21 × 3 → tells us 21 and 3 are factors of 63
But the full list is: 1, 3, 7, 9, 21, 63

The number 7 is a factor of both, but Anshu's two splittings simply did not show it. Guna's splittings (56 = 7 × 8 and 63 = 9 × 7) happened to show it.

The lesson: to be sure about common factors we must break both numbers all the way down to primes:
56 = 2 × 2 × 2 × 7   and   63 = 3 × 3 × 7
Now 7 cannot hide anywhere. 56 and 63 are not co-prime.
Q2.
Try another example: 80 and 63. If we take 80 = 16 × 5 and 63 = 9 × 7, then there are no common factors. Can we conclude that 80 and 63 are co-prime?
Answer

Not from that alone. Anshu's mistake warns us — one lucky splitting proves nothing, because there may be other ways to factorise that reveal a shared factor.

80 = 40 × 2 = 20 × 4 = 10 × 8 = 16 × 5 = ???
63 = 9 × 7 = 3 × 21 = ???

The safe way is to go down to the primes:

80 = 2 × 2 × 2 × 2 × 5
63 = 3 × 3 × 7

The primes of 80 are 2 and 5; the primes of 63 are 3 and 7. Nothing is shared, and now there is nowhere left to hide.

So yes — 80 and 63 really are co-prime, but the prime factorisation is what proves it.

What if they had a composite common factor? Say some composite number c divided both. Then every prime inside c would also divide both — so that prime would show up in both prime factorisations. Since no prime is shared, no common factor bigger than 1 can exist at all.
Q3.
Does the order matter? Using this diagram, can you explain why 30 = 2 × 3 × 5, no matter which way you multiply 2, 3, and 5?
Answer

The picture is a box built out of small cubes, 2 cubes tall, 3 cubes wide and 5 cubes deep. Count the cubes in whichever order you like — the total never changes.

30 = 2 × 3 × 5 — five slabs, each of 2 rows and 3 columns6+6+6+6+66 + 6 + 6 + 6 + 6 = 30
The same 30 cubes, seen as five slabs of 2 rows and 3 columns each.
By slabs → (2 × 3) × 5 = 6 × 5 = 30
By columns → 2 × (3 × 5) = 2 × 15 = 30
Another way → (2 × 5) × 3 = 10 × 3 = 30
And yet another → (3 × 5) × 2 = 15 × 2 = 30

The box does not change when you turn it round, so the count cannot change either.

The two ideas hidden here: we may swap the numbers being multiplied (2 × 3 = 3 × 2) and we may group them any way we like (2 × 3) × 5 = 2 × (3 × 5). In a later class you will meet these as the commutativity and associativity of multiplication. Because of them, the order of the prime factors does not matter — which is why we usually write them in increasing order: 30 = 2 × 3 × 5, 225 = 3 × 3 × 5 × 5.
Q4.
When we find the prime factorisation of a number, we first write it as a product of two factors. For example, 72 = 12 × 6. Then, we find the prime factorisation of each of the factors: 12 = 2 × 2 × 3 and 6 = 2 × 3. Now, can you say what the prime factorisation of 72 is?
Answer

Put the two little factorisations side by side.

72 = 12 × 6
72 = (2 × 2 × 3) × (2 × 3)
72 = 2 × 2 × 3 × 2 × 3

Writing the primes in increasing order:

72 = 2 × 2 × 2 × 3 × 3

Multiply back to check:

2 × 2 × 2 = 8    3 × 3 = 9    8 × 9 = 72 ✔
Tip: it does not matter which first split you choose. Start with 72 = 8 × 9 instead: 8 = 2 × 2 × 2 and 9 = 3 × 3, giving 72 = 2 × 2 × 2 × 3 × 3 — the very same answer.
Q5.
Observe how many times each prime factor occurs in the factorisation of 72. Compare it with how many times it occurs in the factorisations of 12 and 6 put together.
Answer
PrimeTimes in 12 = 2 × 2 × 3Times in 6 = 2 × 3TotalTimes in 72 = 2 × 2 × 2 × 3 × 3
2212 + 1 = 33 ✔
3111 + 1 = 22 ✔

The counts match exactly. So when two numbers are multiplied, the prime factors of the product are simply the prime factors of the two numbers put together.

Why this is so useful: it lets you find the prime factorisation of a big product without ever multiplying it out. For example
56 × 25 = (2 × 2 × 2 × 7) × (5 × 5) = 2 × 2 × 2 × 5 × 5 × 7
You will use exactly this trick in question 4 of the next exercise.
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