1. Meaning of a Square
The square of a number is obtained by multiplying the number by itself.
Key Rule:
Square of n = n × n = n2
Examples:
72 = 7 × 7 = 49
122 = 12 × 12 = 144
252 = 625
Remember:
The expression n2 is read as “n squared”.
2. Perfect Square Numbers
A positive integer is called a perfect square if it can be expressed as the square of an integer.
Definition:
N is a perfect square if:
N = n2
for some integer n.
Examples:
1 = 12
4 = 22
9 = 32
16 = 42
25 = 52
144 = 122
Common Mistake:
A number ending in a possible square digit is not necessarily a perfect square. The unit-digit rule is mainly an elimination test.
3. Important Squares to Remember
Squares from 1 to 30:
12 = 1
22 = 4
32 = 9
42 = 16
52 = 25
62 = 36
72 = 49
82 = 64
92 = 81
102 = 100
112 = 121
122 = 144
132 = 169
142 = 196
152 = 225
162 = 256
172 = 289
182 = 324
192 = 361
202 = 400
212 = 441
222 = 484
232 = 529
242 = 576
252 = 625
262 = 676
272 = 729
282 = 784
292 = 841
302 = 900
Exam Strategy:
Memorising squares at least up to 30 is extremely useful for simplification, square roots, approximation, number system, algebra and quantitative aptitude.
4. Unit Digit of a Perfect Square
Key Property:
The unit digit of a perfect square can only be:
0, 1, 4, 5, 6 or 9.
Therefore:
A perfect square can never end in:
2, 3, 7 or 8.
Examples:
144 ends in 4 → may be a perfect square.
625 ends in 5 → may be a perfect square.
196 ends in 6 → may be a perfect square.
438 ends in 8 → definitely not a perfect square.
Exam Tip:
The unit-digit test can reject impossible options immediately, but it cannot by itself prove that a number is a perfect square.
5. Unit Digit of a Square from the Unit Digit of the Number
Pattern:
If a number ends in 0 → square ends in 0
1 → 1
2 → 4
3 → 9
4 → 6
5 → 5
6 → 6
7 → 9
8 → 4
9 → 1
Example:
A number ending in 7 has a square ending in:
72 = 49
Therefore its square ends in 9.
Useful Pairing:
1 and 9 → square ends in 1
2 and 8 → square ends in 4
3 and 7 → square ends in 9
4 and 6 → square ends in 6
6. Odd and Even Squares
Rule:
Square of an even integer is even.
Square of an odd integer is odd.
Examples:
18 is even → 182 = 324 is even.
17 is odd → 172 = 289 is odd.
Converse for Perfect Squares:
An even perfect square has an even integer square root.
An odd perfect square has an odd integer square root.
7. Prime Factorisation Test for a Perfect Square
Fundamental Rule:
A positive integer is a perfect square if and only if every exponent in its prime factorisation is even.
Example:
3600
= 24 × 32 × 52
All powers are even.
Therefore 3600 is a perfect square.
Example:
180
= 22 × 32 × 5
The exponent of 5 is odd.
Therefore 180 is not a perfect square.
Exam Tip:
This property becomes especially important when finding the least number to multiply or divide by to obtain a perfect square. That application is covered in Topic 5.3.
8. Trailing Zeros in a Perfect Square
Important Property:
A perfect square cannot have an odd number of trailing zeros.
Examples:
100 = 102 → 2 trailing zeros
10,000 = 1002 → 4 trailing zeros
Therefore:
A number ending in exactly 1, 3, 5, ... trailing zeros cannot be a perfect square.
Example:
7840 has one trailing zero.
Therefore it cannot be a perfect square.
9. A Perfect Square Ending in 5
Property:
If a perfect square has unit digit 5, then its last two digits are always 25.
Examples:
52 = 25
152 = 225
252 = 625
352 = 1225
Exam Trap:
Every number ending in 25 is not necessarily a perfect square.
10. Difference Between Consecutive Squares
Formula:
(n+1)2 − n2 = 2n+1
Example:
162 = 256
172 = 289
Difference = 289−256 = 33
Also:
2×16+1 = 33.
Important Pattern:
Successive perfect squares differ by consecutive odd numbers.
11. Numbers Between Two Consecutive Perfect Squares
Rule:
Between n2 and (n+1)2, there are exactly:
2n non-square integers.
Example:
Between 102=100 and 112=121:
Number of integers strictly between them
=121−100−1
=20.
Also 2×10=20.
Exam Tip:
Do not include the two perfect-square endpoints when the question asks for numbers “between” them.
12. Sum of Consecutive Odd Numbers and Squares
Fundamental Identity:
1+3+5+...+(2n−1)=n2
Examples:
1+3=4=22
1+3+5=9=32
1+3+5+7=16=42
Pattern:
The sum of the first n positive odd numbers is always a perfect square.
13. Digital Root Property of Perfect Squares
Useful Test:
For a positive perfect square, the digital root can only be:
1, 4, 7 or 9.
Example:
529 → 5+2+9=16 → 1+6=7.
Since 7 is an allowed digital root, 529 passes this necessary test.
In fact, 529=232.
Important:
This is only a necessary test, not a sufficient test. A number having digital root 1, 4, 7 or 9 is not automatically a perfect square.
14. Quick Identification of a Perfect Square
Fast Checklist:
1. Check the unit digit.
2. Check trailing zeros.
3. Compare with nearby known squares.
4. Use prime-factorisation exponents if needed.
5. For option-based questions, eliminate impossible choices first.
Example:
Is 5329 a perfect square?
722=5184
732=5329
Therefore 5329=732.
15. Common Exam Traps
Trap 1:
Assuming every number ending in 0, 1, 4, 5, 6 or 9 is a perfect square.
Trap 2:
Forgetting that a perfect square cannot end in 2, 3, 7 or 8.
Trap 3:
Ignoring odd prime exponents in prime factorisation.
Trap 4:
Accepting a number with an odd number of trailing zeros as a perfect square.
Trap 5:
Thinking that every number ending in25 is a perfect square.
Trap 6:
Using the digital-root test as final proof instead of an elimination test.
Trap 7:
Including both square endpoints while counting numbers strictly between consecutive squares.
16. Quick Revision
Remember:
• n2=n×n.
• A perfect square is the square of an integer.
• Perfect squares can end only in 0,1,4,5,6 or9.
• They can never end in2,3,7 or8.
• Even integer → even square.
• Odd integer → odd square.
• Every prime exponent of a perfect square is even.
• A perfect square has an even number of trailing zeros.
• A perfect square ending in5 must end in25.
• (n+1)2−n2=2n+1.
• There are2n integers between n2 and(n+1)2.
• Sum of first n odd numbers=n2.
• Digital root of a positive perfect square can only be1,4,7 or9.
• Memorise squares at least up to30.
17. Verified Previous-Year Questions
RRB Technician PYQ
Perfect Square
30 December 2024 · Grade III · Shift 2
Q1. Which of the following numbers is a perfect square?
A. 222
B. 124
C. 141
D. 196
Correct Answer: D. 196
142=14×14=196.
The other options are not perfect squares.
RPF Constable PYQ
Perfect Square
17 January 2019 · Shift 1
Q2. Which of the following is a perfect square?
A. 5329
B. 5327
C. 5322
D. 5328
Correct Answer: A. 5329
732=73×73=5329.
SSC CGL PYQ
Perfect Square Digit
16 August 2017 · Tier-I
Q3. What digit X should replace the unit digit in 211X so that the four-digit number becomes a perfect square?
A. 4
B. 5
C. 6
D. 9
Correct Answer: C. 6
Replacing X by6 gives:
2116=462.
Therefore X=6.
SSC GD PYQ
Square Number Pattern
22 February 2019 · Shift 3
Q4. In the sequence 0, 1, 4, 9, (15), 25, 36, 49, (64), 81, which statement about the bracketed numbers is correct?
A. Both are incorrect
B. Both are correct
C. First is incorrect and second is correct
D. First is correct and second is incorrect
Correct Answer: C
The sequence should contain consecutive squares:
0,1,4,9,16,25,36,49,64,81.
Therefore15 is incorrect, while64 is correct.
18. Practice MCQs
Practice MCQ
Q1. Which number cannot be a perfect square?
A. 625
B. 784
C. 1296
D. 1537
Correct Answer: D. 1537
A perfect square cannot end in7. Therefore1537 cannot be a perfect square.
Practice MCQ
Q2. What is 272?
A. 689
B. 719
C. 729
D. 739
Correct Answer: C. 729
27×27=729.
Practice MCQ
Q3. If a number ends in8, what is the unit digit of its square?
A. 2
B. 4
C. 6
D. 8
Correct Answer: B. 4
82=64, so the square ends in4.
Practice MCQ
Q4. Which prime-factorisation represents a perfect square?
A. 23×32
B. 24×32×52
C. 22×33
D. 2×32×52
Correct Answer: B
Every prime exponent is even:4,2 and2.
Practice MCQ
Q5. How many integers lie strictly between 152 and162?
A. 15
B. 29
C. 30
D. 31
Correct Answer: C. 30
There are2n integers between n² and(n+1)². Here n=15, so2×15=30.
Practice MCQ
Q6. What is the difference between 312 and302?
A. 59
B. 60
C. 61
D. 62
Correct Answer: C. 61
(30+1)²−30²=2×30+1=61.
Practice MCQ
Q7. Find 1+3+5+7+9+11.
A. 30
B. 32
C. 36
D. 42
Correct Answer: C. 36
These are the first6 odd numbers. Their sum=6²=36.
Practice MCQ
Q8. Which of the following cannot be a perfect square because of its trailing zeros?
A. 2500
B. 810000
C. 490000
D. 36000
Correct Answer: D. 36000
36000 has3 trailing zeros. A perfect square cannot have an odd number of trailing zeros.
Practice MCQ
Q9. Which number is a perfect square?
A. 675
B. 676
C. 678
D. 682
Correct Answer: B. 676
26²=676.
Practice MCQ
Q10. If n is an odd integer, which statement is always true?
A. n² is even
B. n² is odd
C. n² ends in5
D. n² is divisible by4
Correct Answer: B. n² is odd
The square of every odd integer is odd.
1. Square अर्थात वर्ग की अवधारणा
किसी number को उसी number से multiply करने पर उसका square या वर्ग प्राप्त होता है।
Key Rule:
n का Square = n × n = n2
उदाहरण:
72 = 7 × 7 = 49
122 = 144
252 = 625
याद रखें:
n2 को “n squared” या “n का वर्ग” पढ़ा जाता है।
2. Perfect Square अर्थात पूर्ण वर्ग
किसी positive integer को perfect square कहा जाता है यदि उसे किसी integer के square के रूप में लिखा जा सके।
Definition:
यदि
N = n2
जहाँ n कोई integer है, तो N एक perfect square है।
उदाहरण:
1=12
4=22
9=32
16=42
25=52
144=122
Common Mistake:
केवल possible unit digit देखकर किसी number को perfect square declare नहीं किया जा सकता। Unit-digit rule मुख्य रूप से elimination के लिए useful है।
3. Important Squares to Remember
1 से30 तक Squares:
12=1, 22=4, 32=9, 42=16, 52=25
62=36, 72=49, 82=64, 92=81, 102=100
112=121, 122=144, 132=169, 142=196, 152=225
162=256, 172=289, 182=324, 192=361, 202=400
212=441, 222=484, 232=529, 242=576, 252=625
262=676, 272=729, 282=784, 292=841, 302=900
Exam Strategy:
कम-से-कम1 से30 तक squares याद रखना simplification, square root, approximation, number system तथा quantitative aptitude में बहुत useful है।
4. Perfect Square का Unit Digit
Key Property:
Perfect square का unit digit केवल:
0, 1, 4, 5, 6 या9
हो सकता है।
इसलिए:
Perfect square कभी भी:
2, 3, 7 या8
पर समाप्त नहीं हो सकता।
Exam Tip:
यह rule impossible options को eliminate करने के लिए बहुत fast है, लेकिन अकेले इससे perfect square prove नहीं होता।
5. Number के Unit Digit से Square का Unit Digit
Pattern:
0 → 0
1 → 1
2 → 4
3 → 9
4 → 6
5 → 5
6 → 6
7 → 9
8 → 4
9 → 1
Useful Pairing:
1 एवं9 → square का unit digit1
2 एवं8 →4
3 एवं7 →9
4 एवं6 →6
6. Odd एवं Even Squares
Rule:
Even integer का square even होता है।
Odd integer का square odd होता है।
उदाहरण:
182=324 → even
172=289 → odd
7. Prime Factorisation द्वारा Perfect Square Test
Fundamental Rule:
किसी positive integer के prime factorisation में प्रत्येक prime की exponent even हो, तभी वह perfect square होगा।
उदाहरण:
3600=24×32×52।
सभी exponents even हैं। इसलिए 3600 perfect square है।
उदाहरण:
180=22×32×5।
5 की exponent1 है, जो odd है। इसलिए180 perfect square नहीं है।
Exam Tip:
Least multiplication/division द्वारा perfect square बनाने वाले questions Topic5.3 में detail में cover होंगे।
8. Perfect Square में Trailing Zeros
Important Property:
Perfect square में trailing zeros की संख्या odd नहीं हो सकती।
उदाहरण:
100 →2 trailing zeros
10000 →4 trailing zeros
इसलिए:
Exactly1,3,5,... trailing zeros वाला integer perfect square नहीं हो सकता।
9. 5 पर समाप्त होने वाला Perfect Square
Property:
यदि perfect square का unit digit5 है, तो उसके last two digits हमेशा 25 होंगे।
उदाहरण:
52=25
152=225
252=625
352=1225
Exam Trap:
हर number जो25 पर समाप्त होता है, perfect square नहीं होता।
10. Consecutive Squares का Difference
Formula:
(n+1)2−n2=2n+1
उदाहरण:
172−162
=289−256
=33।
2×16+1=33।
Pattern:
Consecutive perfect squares का difference consecutive odd numbers होता है।
11. दो Consecutive Perfect Squares के बीच Numbers
Rule:
n2 और(n+1)2 के बीच exactly:
2n integers
होते हैं।
उदाहरण:
100 और121 के बीच:
121−100−1=20 integers हैं।
यह2×10=20 भी है।
12. Odd Numbers का Sum एवं Perfect Square
Identity:
1+3+5+...+(2n−1)=n2
उदाहरण:
1+3=4=22
1+3+5=9=32
1+3+5+7=16=42
13. Perfect Square का Digital Root
Useful Test:
Positive perfect square का digital root केवल:
1, 4, 7 या9
हो सकता है।
Important:
यह केवल elimination test है। Digital root1,4,7 या9 होने से कोई number automatically perfect square नहीं हो जाता।
14. Perfect Square की Quick Identification
Fast Checklist:
1. Unit digit check करें।
2. Trailing zeros check करें।
3. Nearby known squares देखें।
4. जरूरत पर prime factorisation करें।
5. Option-based question में impossible options पहले eliminate करें।
उदाहरण:
5329 के लिए:
732=5329।
इसलिए5329 perfect square है।
15. Common Exam Traps
Trap 1:
0,1,4,5,6 या9 पर ending देखकर सीधे perfect square मान लेना।
Trap 2:
2,3,7 या8 पर ending number को possible perfect square मानना।
Trap 3:
Prime factorisation में odd exponent ignore करना।
Trap 4:
Odd number of trailing zeros वाले number को perfect square मानना।
Trap 5:
25 पर समाप्त हर number को perfect square मानना।
Trap 6:
Digital-root test को final proof मान लेना।
16. Quick Revision
एक नज़र में:
• n2=n×n।
• Perfect square किसी integer का square है।
• Unit digit केवल0,1,4,5,6,9 हो सकता है।
• Perfect square कभी2,3,7,8 पर समाप्त नहीं होता।
• Even number का square even।
• Odd number का square odd।
• Perfect square में सभी prime exponents even होते हैं।
• Trailing zeros की संख्या even होती है।
• 5 पर ending perfect square25 पर समाप्त होता है।
• (n+1)2−n2=2n+1।
• n² और(n+1)² के बीच2n integers होते हैं।
• First n odd numbers का sum=n²।
• Positive perfect square का digital root1,4,7 या9 हो सकता है।
• कम-से-कम1–30 तक squares याद रखें।
17. Verified Previous-Year Questions
RRB Technician PYQPerfect Square30 दिसंबर 2024 · Grade III · Shift 2
प्रश्न 1. निम्न में कौन-सा perfect square है?
A. 222
B. 124
C. 141
D. 196
सही उत्तर: D. 196
142=196।
RPF Constable PYQPerfect Square17 जनवरी 2019 · Shift 1
प्रश्न 2. निम्न में कौन-सा perfect square है?
A. 5329
B. 5327
C. 5322
D. 5328
सही उत्तर: A. 5329
732=5329।
SSC CGL PYQPerfect Square Digit16 अगस्त 2017 · Tier-I
प्रश्न 3. 211X में X के स्थान पर कौन-सा digit रखने पर four-digit number perfect square बनेगा?
A. 4
B. 5
C. 6
D. 9
सही उत्तर: C. 6
X=6 रखने पर2116 मिलता है और462=2116।
SSC GD PYQSquare Number Pattern22 फरवरी 2019 · Shift 3
प्रश्न 4. Sequence 0,1,4,9,(15),25,36,49,(64),81 में bracketed numbers के बारे में सही statement चुनें।
A. दोनों incorrect
B. दोनों correct
C. पहला incorrect, दूसरा correct
D. पहला correct, दूसरा incorrect
सही उत्तर: C
Correct square sequence में15 की जगह16 होना चाहिए, जबकि64=8² correct है। 18. Practice MCQs
Practice MCQ
प्रश्न 1. कौन-सा number perfect square नहीं हो सकता?
A. 625
B. 784
C. 1296
D. 1537
सही उत्तर: D. 1537
Perfect square7 पर समाप्त नहीं हो सकता। Practice MCQ
प्रश्न 2. 272 कितना है?
A. 689
B. 719
C. 729
D. 739
सही उत्तर: C. 729
27×27=729। Practice MCQ
प्रश्न 3. किसी number का unit digit8 हो तो उसके square का unit digit क्या होगा?
A. 2
B. 4
C. 6
D. 8
सही उत्तर: B. 4
8²=64। इसलिए unit digit4 होगा। Practice MCQ
प्रश्न 4. कौन-सा prime factorisation perfect square represent करता है?
A. 23×32
B. 24×32×52
C. 22×33
D. 2×32×52
सही उत्तर: B
सभी prime exponents even हैं। Practice MCQ
प्रश्न 5. 15² एवं16² के strictly बीच कितने integers हैं?
A. 15
B. 29
C. 30
D. 31
सही उत्तर: C. 30
2n=2×15=30। Practice MCQ
प्रश्न 6. 31² एवं30² का difference कितना है?
A. 59
B. 60
C. 61
D. 62
सही उत्तर: C. 61
2×30+1=61। Practice MCQ
प्रश्न 7. 1+3+5+7+9+11 का sum क्या है?
A. 30
B. 32
C. 36
D. 42
सही उत्तर: C. 36
First6 odd numbers का sum=6²=36। Practice MCQ
प्रश्न 8. Trailing zeros के आधार पर कौन-सा number perfect square नहीं हो सकता?
A. 2500
B. 810000
C. 490000
D. 36000
सही उत्तर: D. 36000
इसमें3 trailing zeros हैं, जो odd संख्या है। Practice MCQ
प्रश्न 9. निम्न में perfect square कौन-सा है?
A. 675
B. 676
C. 678
D. 682
सही उत्तर: B. 676
26²=676। Practice MCQ
प्रश्न 10. यदि n odd integer है, तो कौन-सा statement हमेशा true है?
A. n² even है
B. n² odd है
C. n² का unit digit5 है
D. n² हमेशा4 से divisible है
सही उत्तर: B. n² odd है
हर odd integer का square odd होता है।