NCERT Solutions Ganita Prakash Chapter 5 In-text Questions — Fun with numbers

Book page 126 to 128 Updated on2026-09-05

Q1.

There are four numbers in this box — 9, 16, 25, 43. Which number looks special to you? Why do you say so?

Answer

Every one of the four can be called special — it depends on what you look at. Here are the reasons Guna's classmates gave, and a few more.

Number	Why it is special
9	The only single-digit number. The only multiple of 3. The only number here that is smaller than 10.
16	The only even number. The only multiple of 4. The only power of 2. The only number here with five factors (1, 2, 4, 8, 16).
25	The only multiple of 5. The only number that ends in 5.
43	The only prime number. The only number that is not a perfect square (9 = 3², 16 = 4², 25 = 5²).

The point of the question: there is no single “correct” answer. Any property that exactly one of the four numbers has makes that number special. The fun is in finding a reason that nobody else in the class thought of.

Q2.

Below are some boxes with four numbers in each box. Within each box try to say how each number is special compared to the rest. Share with your classmates and find out who else gave the same reasons as you did. Did anyone give different reasons that may not have occurred to you?

Answer

Box 1 — 5, 7, 12, 35

5 — the smallest number here; the only one that is a factor of 25; also the only number that appears as a digit inside another number of the box (the 5 of 35).
7 — the only number that is a factor of 49; also the only one whose digit does not appear anywhere else in the box.
12 — the only even number; the only multiple of 3; the only one that is not a factor of 35.
35 — the only odd composite number; and the only number that is the product of two others in the box (35 = 5 × 7).

Box 2 — 3, 8, 11, 24

3 — the only odd single-digit number; the smallest number here.
8 — the only perfect cube (8 = 2 × 2 × 2); also the only power of 2.
11 — the only number that is not a factor of 24; also the only two-digit prime here.
24 — the largest number; the only one with more than four factors; and the product of two others (24 = 3 × 8).

Box 3 — 27, 3, 123, 31

27 — the only perfect cube (3 × 3 × 3); also the only multiple of 9.
3 — the only single-digit number; also a factor of both 27 and 123.
123 — the only three-digit number; the only one whose digits are three different numbers in a row.
31 — the only number that is not a multiple of 3 (27, 3 and 123 all are, since 1 + 2 + 3 = 6).

Box 4 — 17, 27, 44, 65

17 — the only prime number.
27 — the only multiple of 3; also the only perfect cube.
44 — the only even number; also the only one with a repeated digit.
65 — the only multiple of 5; the largest number here.

Math Talk: compare your list with a friend's. Good properties to hunt for are — odd or even, prime or composite, a square or a cube, a multiple of 3 or 5, the number of digits, and the number of factors.

Q3.

A prime puzzle. The figure on the left shows the puzzle. The figure on the right shows the solution of the puzzle. Think what the rules can be to solve the puzzle.

Answer

Comparing the two figures gives the rules at once:

The rules: fill the 3 × 3 grid with prime numbers only, so that

the product of the three numbers in each row equals the number written to the right of that row, and
the product of the three numbers in each column equals the number written below that column.

Check the solved example given in the book:

	Col 1	Col 2	Col 3	Row product
Row 1	5	5	3	5 × 5 × 3 = 75 ✔
Row 2	2	3	7	2 × 3 × 7 = 42 ✔
Row 3	17	2	3	17 × 2 × 3 = 102 ✔
Column product	5 × 2 × 17 = 170 ✔	5 × 3 × 2 = 30 ✔	3 × 7 × 3 = 63 ✔

How to start solving one: prime factorise every row total and every column total. If a total is a product of exactly three primes — like 102 = 2 × 3 × 17 — that row or column is fixed at once, and the rest follows. A useful check: the product of all three row totals must equal the product of all three column totals, since both count the same nine numbers.

Q4.

Rules: Fill the grid with prime numbers only so that the product of each row is the number to the right of the row and the product of each column is the number below the column. (Four puzzles are given on pages 127 and 128.)

Answer

Prime factorise every total first, then look for a row or column that is forced.

Puzzle 1 — rows 105, 20, 30; columns 28, 125, 18

125 = 5 × 5 × 5 → the whole middle column must be 5, 5, 5
Row 2: ? × 5 × ? = 20 → the other two multiply to 4 → both are 2
Column 3: ? × 2 × ? = 18 → the other two multiply to 9 → both are 3
Row 1: ? × 5 × 3 = 105 → first entry = 7 Row 3: ? × 5 × 3 = 30 → first entry = 2

	Col 1	Col 2	Col 3	Row product
Row 1	7	5	3	105
Row 2	2	5	2	20
Row 3	2	5	3	30
Column	7 × 2 × 2 = 28	5 × 5 × 5 = 125	3 × 2 × 3 = 18

Puzzle 2 — rows 8, 105, 70; columns 30, 70, 28

8 = 2 × 2 × 2 → the whole first row must be 2, 2, 2
Then column 1: 2 × ? × ? = 30 → the rest multiply to 15; column 2 → 35; column 3 → 14
Row 3 = 70 = 2 × 5 × 7, and its 2 can only sit in column 3 (14 = 2 × 7)

	Col 1	Col 2	Col 3	Row product
Row 1	2	2	2	8
Row 2	3	5	7	105
Row 3	5	7	2	70
Column	2 × 3 × 5 = 30	2 × 5 × 7 = 70	2 × 7 × 2 = 28

Puzzle 3 — rows 63, 27, 190; columns 45, 42, 171

27 = 3 × 3 × 3 → the whole middle row must be 3, 3, 3
190 = 2 × 5 × 19 has no 3 in it, and 171 = 3 × 3 × 19 → the 19 must be in row 3, column 3

	Col 1	Col 2	Col 3	Row product
Row 1	3	7	3	63
Row 2	3	3	3	27
Row 3	5	2	19	190
Column	3 × 3 × 5 = 45	7 × 3 × 2 = 42	3 × 3 × 19 = 171

Puzzle 4 — rows 343, 66, 44; columns 28, 154, 231

343 = 7 × 7 × 7 → the whole first row must be 7, 7, 7
Column 1: 7 × ? × ? = 28 → the rest multiply to 4 → both are 2
Row 2: 2 × ? × ? = 66 → the rest multiply to 33; Row 3: 2 × ? × ? = 44 → the rest multiply to 22

	Col 1	Col 2	Col 3	Row product
Row 1	7	7	7	343
Row 2	2	11	3	66
Row 3	2	2	11	44
Column	7 × 2 × 2 = 28	7 × 11 × 2 = 154	7 × 3 × 11 = 231

Check every puzzle this way: multiply the three row totals and the three column totals — they must agree. Puzzle 1: 105 × 20 × 30 = 63000 and 28 × 125 × 18 = 63000 ✔. Puzzle 4: 343 × 66 × 44 = 996072 and 28 × 154 × 231 = 996072 ✔.

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