Exercise 6.1 — Square Properties

Problems based on properties of square numbers.

Lesson Notes PDF

1 / —

Loading PDF…

Exercise 6.1 – Units Digit Patterns, Perfect Squares & Odd Number Sums

Exercise 6.1 of Class 8 Mathematics (CBSE, Telangana & Andhra Pradesh syllabus) puts the patterns from the chapter introduction into practice. Every question here is solved without performing the full multiplication — instead, you apply pattern-recognition rules about units digits, even/odd behaviour, and the relationships between consecutive squares and odd-number sums.

This exercise has six parts covering: finding the units digit of a square, identifying perfect squares, explaining why a number isn't a perfect square, determining whether a square is even or odd, counting integers between consecutive squares, and finding sums of consecutive odd numbers using the n² shortcut.

💡 The big idea of Exercise 6.1: You never need to actually compute the square of a large number to answer these questions. Every answer comes from a pattern rule — recognising which rule applies is the entire skill being tested.

Question 1

Finding the Units Digit of a Square Without Squaring the Whole Number

Units Digit Rule

To find the units digit of a large number's square, you only need to look at the units digit of the original number and square that single digit. The units digit of the result becomes your answer — the rest of the number doesn't matter at all.

Units digit of n² depends only on the units digit of n

Number	Units Digit of Number	Square of That Digit	Units Digit of n²
(i) 39	9	9² = 81	1
(ii) 297	7	7² = 49	9
(iii) 5125	5	5² = 25	5
(iv) 7286	6	6² = 36	6
(v) 8742	2	2² = 4	4

Example: For 297 — its units digit is 7
7² = 49, whose units digit is 9
∴ Units digit of 297² = 9

📐 Notice that digits 1, 5, 6, and 0 always "reproduce themselves" in the units place when squared (1² = 1, 5² = 25, 6² = 36, 0² = 0) — a useful shortcut to remember for instant recall.

Question 2

Identifying Perfect Squares Using the Units Digit Filter

Perfect Square Check

For each number, we first check the units digit. If it's 2, 3, 7, or 8, the number is immediately ruled out as a perfect square. If the units digit passes this test (0, 1, 4, 5, 6, or 9), we then check against the squares table from the chapter introduction (1² to 30²) to confirm.

Number	Units Digit	Verdict	Reason
(i) 121	1	Perfect square	121 = 11²
(ii) 136	6	Not a perfect square	Passes units-digit test, but 136 doesn't appear in the squares table (11² = 121, 12² = 144)
(iii) 256	6	Perfect square	256 = 16²
(iv) 321	1	Not a perfect square	Passes units-digit test, but lies between 17² = 289 and 18² = 324
(v) 600	0	Not a perfect square	Ends in only one zero — squares of multiples of 10 must end in two zeroes

💡 Two-step check: First eliminate numbers ending in 2, 3, 7, or 8 instantly. For the remaining numbers, compare against the nearest perfect squares above and below to confirm — a number that falls strictly between two consecutive perfect squares cannot itself be a perfect square.

Question 3

Explaining Why Given Numbers Cannot Be Perfect Squares

Reasoning — Units Digit & Trailing Zeroes

This question asks for the reasoning, not just a yes/no answer. Every explanation here uses one of two rules: either the units digit doesn't belong to the set {0, 1, 4, 5, 6, 9}, or the number ends in an odd number of zeroes.

Number	Units Digit / Ending	Reason It's Not a Perfect Square
(i) 257	7	Units digit 7 — square numbers never end in 7
(ii) 4592	2	Units digit 2 — square numbers never end in 2
(iii) 2433	3	Units digit 3 — square numbers never end in 3
(iv) 5050	ends with one zero	Squares of multiples of 10 must end with two zeroes, not one
(v) 6098	8	Units digit 8 — square numbers never end in 8

📐 Board exam tip: When a question asks "give reasons," writing just "not a perfect square" without explanation loses marks. Always state which rule (units digit set, or trailing-zeroes rule) the number violates.

Question 4

Determining Whether the Square of a Number Is Even or Odd

Even/Odd Square Rule

This is one of the simplest rules in the chapter: the square of an even number is always even, and the square of an odd number is always odd. You don't need to calculate the square at all — just check whether the original number is even or odd.

Number	Even / Odd	Square Is
(i) 431	Odd	Odd
(ii) 2826	Even	Even
(iii) 8204	Even	Even
(iv) 17779	Odd	Odd
(v) 99998	Even	Even

Even number → Even square | Odd number → Odd square

Question 5

Counting Integers Between the Squares of Consecutive Numbers

2n Rule

This question applies the formula from the chapter introduction directly: the number of integers lying between n² and (n+1)² is always 2n. Simply take the smaller number, double it, and that's your answer — no need to calculate either square.

Number of integers between n² and (n + 1)² = 2n

Between Squares Of	Calculation (2n)	Number of Integers
(i) 25 and 26	2 × 25	50
(ii) 56 and 57	2 × 56	112
(iii) 107 and 108	2 × 107	214

Example: Between 25² and 26²
Number of integers = 2 × 25
= 50

Question 6

Finding the Sum of Consecutive Odd Numbers Without Adding

Sum of Odd Numbers = n²

Instead of adding a long list of odd numbers one by one, simply count how many odd numbers are in the list (that's your value of n), and the sum is n². Since odd numbers always start from 1 in this pattern, the count directly gives the base number to be squared.

Sum of first n odd natural numbers = n²

Sum	Count of Terms (n)	Result
(i) 1 + 3 + 5 + 7 + 9	5	5² = 25
(ii) 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17	9	9² = 81
(iii) 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21 + 23 + 25	13	13² = 169

Example: 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17
Count of terms = 9 (since these are the first 9 odd numbers starting from 1)
Sum = 9²
= 81

💡 Quick counting tip: For a sequence of odd numbers starting at 1 and ending at a number k, the count n = (k + 1) ÷ 2. For example, the last term 25 gives n = (25+1)/2 = 13, confirming the sum is 13² = 169.

Common Mistakes to Avoid in Exercise 6.1

Squaring the entire number instead of just the units digit: For Q1, you only need to square the last digit of the number — squaring the whole number wastes time and increases error risk.
Treating the units-digit rule as a guarantee: A number ending in 0, 1, 4, 5, 6, or 9 is not automatically a perfect square (see 136 and 321 in Q2) — it only passes the first filter. Always verify against known squares too.
Confusing "ends in zero" with "ends in two zeroes": For Q3(iv), 5050 ends in just one zero. Perfect squares of multiples of 10 must end in exactly two zeroes (like 100, 400, 900).
Miscounting terms in odd-number sums: For Q6, carefully count how many terms are in the sequence — an off-by-one error here directly changes the value of n and therefore the final answer.
Forgetting the "strictly between" condition: In Q5, the 2n formula counts integers strictly between the two squares — don't include the squares themselves.

📐 Board exam tip: Across CBSE, Telangana, and Andhra Pradesh exams, questions like Q3 specifically test whether you can explain a rule, not just apply it. Practise writing one-line justifications using the exact pattern names: "units digit rule" or "trailing zeroes rule."

What Exercise 6.1 Prepares You For

The pattern-recognition skills built in this exercise — especially the units digit rule and perfect square identification — are essential for the next part of this chapter, where you'll learn to find the square root of a number using prime factorisation and the long division method. Being able to quickly judge whether a number is a perfect square saves significant time when checking your final answers.

These ideas also build directly on the foundational concepts covered in the Introduction to Square Roots and Cube Roots, and they connect with Exponents and Powers, where the same n² notation is explored with a wider range of exponents and bases.