Squares and Square Roots Chapter 5 solution

Squares and Square Roots Chapter 5 solution

• If a natural number m can be expressed as n², where n is also a natural number, then m is a square number.
• The numbers 1, 4, 9, 16 … are square numbers. These numbers are also called perfect squares.

Properties of Square Numbers

Number	Square
1	1
2	4
3	9
4	16
5	25
6	36
7	49
8	64
9	81
10	100
11	121
12	144
13	169
14	196
15	225
16	256
17	289
18	324
19	361
20	400

• What are the ending digits (that is, digits in the units place) of the square numbers?
• All these numbers end with 0, 1, 4, 5, 6 or 9 at units place. None of these end with 2, 3, 7 or 8 at unit’s place.

Question :

Quwstion :Write five numbers which you cannot decide just by looking at their units digit (or units place) whether they are square numbers or not.

Solution : Here are five numbers where you cannot decide just by looking at their unit digit whether they are square numbers or not:

1. 21
2. 31
3. 61
4. 81
5. 91

Explanation:

Some numbers like 4, 9, 16, 25, 36, 49, 64, 81, 100 are clearly square numbers, and their units digits (0, 1, 4, 5, 6, or 9) can often help you identify a possible square.

However, some non-square numbers also end with those digits. For example:

• 21 ends with 1, but it’s not a square number.
• 81 also ends with 1, but it is a square number (9²).
• So, just seeing the dig
• it 1 at the end doesn’t help you decide for sure.

Resource : https://www.ncertpdf.com/

Note : if a number has 1 or 9 in the units place, then it’s square ends in 1.

Question :

Solution :

• 123² = 15129 = Unit digit is 9
• 77² = 5929 = Unit digit is 9
• 82² = 6727 = Unit digit is 7
• 161² = 25921 = Unit digit is 1
• 109² = 11881 = Unit digit is 1

Question :

Solution :

• 19² = 361 = Unit digit is 1
• 24² = 576 = Unit digit is 6
• 26² = 676 = Unit digit is 6
• 36² = 1296 = Unit digit is 6
• 34² = 1156 =Unit digit is 6

hence , 24² , 26² ,36² , and 34² has unit digit 6 Ans.

Note : when a square number ends in 6, the number whose square it is, will have either 4 or 6 in unit’s place.

Question :

Solution :

1. 1234 = the one’s digit (units digit) of the square of 1234, we only need to look at the one’s digit of 1234, which is 4.

Now, square the digit 4: 4²=16

So, the one’s digit of 1234² is 6. ✅

II. 26387 = the one’s digit of the square of 26387, follow these steps:

Look at the one’s digit of 26387:
It is 7.

Square the digit 7:

7²=49

The one’s digit of 49 is 9.

✅ Final Answer: The one’s digit of 26387² is 9.

III. 52698 :

the one’s digit of the square of 52698:

Look at the one’s digit of 52698:
It is 8.

Square the digit 8:

8²= 64

The one’s digit of 64 is 4.

✅ Final Answer: The one’s digit of 52698² is 4.

IV. 99880 = the one’s digit of the square of 99880:

Look at the one’s digit of 99880:
It is 0.

Square the digit 0:

0²=0

✅ Final Answer: The one’s digit of 99880² is 0.

21222

• One’s digit of 21222 is 2
• 2²=4
• ✅ So, the one’s digit of 21222² is 4

⁹¹⁰⁶

• One’s digit of 9106 is 6
• 6²=36
• ✅ So, the one’s digit of 9106² is 6

✅ Final Answers:

• 21222² → One’s digit: 4
• 9106² → One’s digit: 6

Note :

Resourse : NCERT text book

NOTE :

• The square of an odd number is always odd.
• The square of an even number is always even.

Question :

Solution :

I. 727

• Ends in 7 → Odd number
• ✅ Square of 727 will be odd
• Reason: Odd × Odd = Odd

II. 158

• Ends in 8 → Even number
• ✅ Square of 158 will be even
• Reason: Even × Even = Even

III. 269

• Ends in 9 → Odd number
• ✅ Square of 269 will be odd
• Reason: Odd × Odd = Odd

IV. 1980

• Ends in 0 → Even number
• ✅ Square of 1980 will be even
• Reason: Even × Even = Even

✅ Final Answers:

Number	Odd/Even	Square Will Be	Reason
727	Odd	Odd	Odd × Odd = Odd
158	Even	Even	Even × Even = Even
269	Odd	Odd	Odd × Odd = Odd
1980	Even	Even	Even × Even = Even

(i) 60

60²=3600

• ✅ Number of trailing zeros = 2

(ii) 400

400²=160000

• ✅ Number of trailing zeros = 4

✅ Final Answers:

Number	Square	Trailing Zeros
60	3600	2
400	160000	4

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NOTE : There are 2n non perfect square numbers between the squares of the numbers n and (n + 1)

Question :

Solution :🔹 Formula:

If you are given two numbers, say a and b such that a<b then: Number of natural numbers between a² and b²= ( b²−a²−1 )

i) Between 9² and 10²

9²=81,10²=100

Natural numbers between 81 and 100=100−81−1=18​

(ii) Between 11² and 12²

11²=121,12²=144 Natural numbers between 121 and 144=144−121−1=22

✅ Final Answers:

• Between 9² and 10² → 18 natural numbers
• Between 11² and 12² → 22 natural numbers

NOTE :

(i) Between 100² and 101²

100²=10,000,101²=10,201

Total numbers between=10,201−10,000−1=200

Since there are no perfect squares between these two,
Number of non-square numbers = 200

(ii) Between 90² and 91²

90²=8,100,91²=8,

Total numbers between=8,281−8,100−1=180

Again, no perfect squares between these two,
Number of non-square numbers = 180

✅ Final Answers:

Pair	Non-square numbers between
1002and 1012	200
902and 912	180

Some More Interesting Patterns

• Adding triangular numbers.
• Numbers between square numbers : that there are 2n non perfect square numbers between the squares of the numbers n and (n + 1)
• Adding odd numbers : the sum of first n odd natural numbers is n². if a natural number cannot be expressed as a sum of successive odd natural numbers starting with 1, then it is not a perfect square.
• A sum of consecutive natural numbers : The square of any odd number as the sum of two consecutive positive integers.

• Product of two consecutive even or odd natural numbers

• Some more patterns in square numbers : if a natural number cannot be expressed as a sum of successive odd natural numbers starting with 1, then it is not a perfect square.

EXERCISE 5.1

Solution :

Number	Unit Digit	Square of Unit Digit	Unit Digit of the Square
81	1	1² = 1	1
272	2	2² = 4	4
799	9	9² = 81	1
3853	3	3² = 9	9
1234	4	4² = 16	6
26387	7	7² = 49	9
52698	8	8² = 64	4
99880	0	0² = 0	0
12796	6	6² = 36	6
55555	5	5² = 25	5

Solution :

Number	Last Digit(s)	Rule Violated / Reason	Conclusion
1057	7	Perfect squares cannot end in 7	❌ Not a perfect square
23453	3	Perfect squares cannot end in 3	❌ Not a perfect square
7928	8	Perfect squares cannot end in 8	❌ Not a perfect square
222222	2	Perfect squares cannot end in 2	❌ Not a perfect square
64000	Ends in 3 zeros	Perfect squares must end in even number of zeros	❌ Not a perfect square
89722	2	Perfect squares cannot end in 2	❌ Not a perfect square
89722	2	Same as above	❌ Not a perfect square
222000	Ends in 3 zeros	Odd number of zeros → not a perfect square	❌ Not a perfect square
505050	Ends in 50	No perfect square ends in 50	❌ Not a perfect square

Solution :

Number	Last Digit	Odd/Even	Square Will Be
431	1	Odd	✅ Odd
2826	6	Even	❌ Even
7779	9	Odd	✅ Odd
82004	4	Even	❌ Even

Question :

(i)Express 49 as the sum of 7 odd numbers. (ii) Express 121 as the sum of 11 odd numbers.

Question :

How many numbers lie between squares of the following numbers? (i) 12 and 13 (ii) 25 and 26 (iii) 99 and 100

Solution :