NCERT Solutions Ganita Prakash (Part 1) Chapter 5 .1 Is This a Multiple Of? — In-text Questions

Book page 1145 Updated on2026-09-05

Q1.

Now take any 4 numbers, place ‘+’ and ‘–’ signs in the eight different ways, and evaluate the resulting expression. What do you observe about their parities? Repeat this with other sets of 4 numbers.

Answer

All eight expressions have the same parity — and it is the parity of a + b + c + d.

Four numbers	a + b + c + d	a + b – c – d	a – b – c – d	Parity of all 8
2, 5, 7, 4	18	– 4	– 14	even
1, 2, 3, 5	11	– 5	– 9	odd
10, 4, 6, 8	28	0	– 8	even
3, 3, 3, 3	12	0	– 6	even

With four consecutive numbers two of them are odd and two are even, so a + b + c + d is always even — that is the special case we met on the last page. With any four numbers the shared parity can be odd instead, but it is still shared by all eight.

Q2.

Is there a way to explain why this happens? Hint: Think of the rules for parity of the sum or difference of two numbers.

Answer

Yes. The key is that switching one sign changes the value by an even number, so it can never change the parity.

Take a + b – c – d and replace + b by – b:
(a + b – c – d) – (a – b – c – d)
= a + b – c – d – a + b + c + d
= 2b — an even number

If two numbers differ by an even number, they cannot have different parities — either both are even or both are odd.

Why it happens: starting from any one of the eight expressions you can reach the other seven by switching one, two or three signs. Every switch changes the value by 2 × (one of the numbers), which is even. A chain of even changes is still an even change, so all eight expressions share one parity.

The second explanation in the book reaches the same place through the parity rules:

odd ± odd = even even ± even = even odd ± even = odd

In every rule the '+' case and the '–' case give the same answer. So a ± b has one parity whichever sign is used; then a ± b ± c has one parity; then a ± b ± c ± d does too.

Q3.

Replace any negative sign in the expression a + b – c – d with a positive sign and find the difference between the two numbers. What do you conclude from this observation?

Answer

Change – c to + c:

(a + b + c – d) – (a + b – c – d)
= a + b + c – d – a – b + c + d
= 2c — an even number

Change – d to + d instead and the difference is 2d, again even.

Conclusion: it makes no difference which sign you switch or which way you switch it. Turning + into – changes the value by – 2 × (that number); turning – into + changes it by + 2 × (that number). Both are even. So the parity of the expression is fixed the moment the four numbers are chosen — the signs cannot touch it.

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