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NCERT Solutions for Class 5th Maths Chapter 13 The multiplying box; factors, multiples and arrays — Find the Hidden Numbers

Book page 164 Updated on2026-09-19

Q1.
(a) Can you guess the multiplier if you see the 4 numbers coming out of the box?
Answer

The multiplier can be 1, 2 or 4.

The box does the same thing to every number put in — it multiplies it. So the multiplier must fit exactly into all four numbers that came out: 28, 36, 48 and 72.

  1. List every number that divides 28 exactly. 1, 2, 4, 7, 14, 28.
  2. Do the same for 36. 1, 2, 3, 4, 6, 9, 12, 18, 36.
  3. Do the same for 48. 1, 2, 3, 4, 6, 8, 12, 16, 24, 48.
  4. Do the same for 72. 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72.
  5. Tick the numbers that are in all four lists. Only 1, 2 and 4 are in every list.
28 = 4 × 7
36 = 4 × 9
48 = 4 × 12
72 = 4 × 18

So the multiplier can be 4 (and also 2, or 1)
Why it happens: A number that divides another number exactly, with nothing left over, is called a factor. The multiplier had to be a factor of every number that came out, because the box built each of those numbers by multiplying. So the answer is the list of factors that 28, 36, 48 and 72 all share.
Tip: 4 is the most interesting guess, because it is the biggest of the three. A multiplier of 1 would mean the box did nothing at all.
Q2.
(b) Is there more than one possible multiplier?
Answer

Yes. There are three possible multipliers: 1, 2 and 4.

MultiplierDoes it divide 28, 36, 48 and 72 exactly?
1Yes — 1 divides every number
2Yes — all four numbers are even
4Yes — 28 = 4 × 7, 36 = 4 × 9, 48 = 4 × 12, 72 = 4 × 18
3No — 28 ÷ 3 leaves 1 over
6No — 28 ÷ 6 leaves 4 over
Why it happens: 1, 2 and 4 are the common factors of 28, 36, 48 and 72 — the factors that all four numbers share. Any one of them could have been the multiplier.
Check it yourself: Try 8. 48 ÷ 8 = 6 and 72 ÷ 8 = 9, but 28 ÷ 8 leaves 4 over. So 8 fails, and it is not a common factor.
Q3.
(c) What numbers might have been put inside the box?
Answer

It depends on which multiplier the box was using. Here are all three sets.

If the multiplier is…Then the numbers put in were…Working
47, 9, 12, 1828 ÷ 4 = 7, 36 ÷ 4 = 9, 48 ÷ 4 = 12, 72 ÷ 4 = 18
214, 18, 24, 3628 ÷ 2 = 14, 36 ÷ 2 = 18, 48 ÷ 2 = 24, 72 ÷ 2 = 36
128, 36, 48, 72Nothing changes when you multiply by 1
To go backwards, divide.
Number that came out ÷ multiplier = number that went in
28 ÷ 4 = 7
Why it happens: Multiplying and dividing undo each other. The box multiplied, so to find what went in you divide.
Look at the picture again: Faint numbers are drawn near the box in your book — 12, 7 and 18 are three of them. All three are in the answer for a multiplier of 4.
Q4.
Share your thoughts in the class. You can also play this game with your friends.
Answer

Sample answer: The box must be multiplying by 4. All four numbers that came out — 28, 36, 48 and 72 — can be split into equal groups of 4 with nothing left over. So the numbers that went in were 7, 9, 12 and 18. A multiplier of 2 also works, and even 1 works, but 1 would mean the box changed nothing.

How to play it with your friends, step by step:

  1. One player is the box. That player secretly chooses a multiplier, say 6.
  2. The others call out numbers. Say they call out 3, 5 and 8.
  3. The box answers. The box says 18, 30 and 48 — each number multiplied by 6.
  4. Guess the multiplier. Look for the biggest number that divides 18, 30 and 48 exactly. It is 6.
  5. Swap roles and play again.
Try This: Make the game harder by choosing a multiplier like 7 or 9. The numbers that come out grow fast, and your friends must hunt for factors instead of guessing.
Q5.
A number, when arranged in an array, shows the factors of that number. Are there other numbers that are factors of 15? Try to make other arrays for the number 15.
Answer

Yes. Besides 3 and 5, the numbers 1 and 15 are also factors of 15. So 15 has four factors in all: 1, 3, 5 and 15.

An array means the counters set out in equal rows, in a rectangle. Each rectangle you can make gives you a pair of factors.

3 × 5 = 155 × 3 = 15
Two rectangles for 15. Both use 15 counters — one is the other turned round.
1 × 15 = 15
A third rectangle: one single row of 15. This one gives the factors 1 and 15.
3 × 5 = 15 → factors 3 and 5
5 × 3 = 15 → the same two factors
1 × 15 = 15 → factors 1 and 15

Factors of 15 = 1, 3, 5, 15
Why it happens: If the counters fit into a neat rectangle with no gaps, then the number of rows divides the total exactly. That is exactly what a factor is. A rectangle you cannot make — say 4 rows of something — means 4 is not a factor of 15.
Check it yourself: Try 2 rows. 15 counters cannot be split into 2 equal rows — one row would be 7 and the other 8. So 2 is not a factor of 15.
Q6.
Let us make arrays for the number 12. (The book shows 3 × 4 = 12, 2 × 6 = 12 and 1 × 12 = 12.) Which numbers are the factors of 12?
Answer

The factors of 12 are 1, 2, 3, 4, 6 and 12 — six factors in all.

3 × 4 = 122 × 6 = 12
Two of the rectangles the book draws for 12.
1 × 12 = 12
The third rectangle: one long row of 12.

Every rectangle hands you a pair of factors. Collect the pairs:

1 × 12 = 12 → 1 and 12
2 × 6 = 12 → 2 and 6
3 × 4 = 12 → 3 and 4

Factors of 12 = 1, 2, 3, 4, 6, 12
Why it happens: Turning a rectangle on its side gives the same pair of factors. 3 rows of 4 and 4 rows of 3 both use 12 counters, so they tell us the same thing. That is why we only need three rectangles to find all six factors.
Tip: To hunt for factors, try 1, then 2, then 3, then 4 … and stop when the numbers start repeating. For 12, after 3 × 4 the next try is 4 × 3, which we already have — so you can stop.
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