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Properties of Square Numbers Quiz - CBSE Class 8 Mathematics

Free practice quiz: Properties of Square Numbers (CBSE Class 8 Mathematics). 10 multiple-choice questions with answers and explanations. Play online, no sign-up needed.

This CBSE Class 8 Mathematics practice quiz has 10 multiple-choice questions on Properties of Square Numbers, and 10 of them come with an explanation. Try the 5 sample questions below (tap an option to check yourself), then play the full quiz online. No sign-up needed.

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Units digit of a square

  • The units digit of a square depends only on the units digit of the original number.
  • Square just the last digit; the last digit of that result is the answer.
  • A perfect square never ends in 3, 7 or 8, so such numbers are not squares.
Ends in 3 or 7
\[n \equiv 3 \text{ or } 7 \pmod{10} \Rightarrow n^2 \equiv 9 \pmod{10}\]
Ends in 5
\[n \equiv 5 \pmod{10} \Rightarrow n^2 \equiv 5 \pmod{10}\]
Ends in 6
\[n \equiv 6 \pmod{10} \Rightarrow n^2 \equiv 6 \pmod{10}\]

Common mistake: Thinking a number is a perfect square just because its last digit is allowed. A suitable last digit does not guarantee a square.

Zeros at the end

  • If a number ends in z zeros, its square ends in twice as many zeros.
  • So a perfect square can only end in an even number of zeros.
Zeros double when squaring
\[(a \times 10^{z})^2 = a^2 \times 10^{2z}\]

Common mistake: Counting the zeros of the original number and stopping there, instead of doubling them for the square.

Even and odd squares

  • The square of an even number is even, and the square of an odd number is odd.
  • The sum of two odd numbers is always even; the sum of two even numbers is also even.
Even squared
\[(2m)^2 = 4m^2\]
Odd squared
\[(2m+1)^2 = 4m^2 + 4m + 1\]
Product of consecutive integers is even
\[m(m+1) \text{ is even}\]

Common mistake: Assuming an odd number squared can be even. Odd times odd is always odd.

Squares and odd numbers

  • The difference of two consecutive squares equals the sum of the two numbers.
  • Adding the first m odd numbers always gives exactly m squared.
Consecutive squares
\[(a+1)^2 - a^2 = a + (a+1)\]
Sum of first m odd numbers
\[1 + 3 + 5 + \cdots + (2m-1) = m^2\]

Common mistake: Mixing up which number is the smaller one when matching the difference of consecutive squares to a formula.

Exam tips

Beta: made with AI and checked by a second AI. A teacher has not reviewed it yet.

Sample questions

  1. Which of the following numbers is a perfect square?

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    Answer: C. 25

    25 is a perfect square because 5 x 5 = 25.

  2. If a number ends in 1, its square can end in which of the following digits?

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    Answer: C. 1, 6

    1 x 1 = 1; 11 x 11 = 121; 21 x 21 = 441; 31 x 31 = 961; 41 x 41 = 1681; The pattern of ones digit is 1, 1, 1, 1, 1...; 91 x 91 = 8281. If it ends in 9, the square ends in 1. If it ends in 1, the square ends in 1.

  3. Can a perfect square end in 2?

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    Answer: B. No

    The ones digit of a perfect square can only be 0, 1, 4, 5, 6, or 9.

  4. The sum of two consecutive odd numbers is always:

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    Answer: B. Even

    Let the two consecutive odd numbers be 2n+1 and 2n+3. Their sum is (2n+1) + (2n+3) = 4n+4 = 2(2n+2), which is always even.

  5. If the difference between two consecutive squares is 2n+1, then n is?

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    Answer: C. Any number

    Let the consecutive squares be n² and (n+1)². The difference is (n+1)² - n² = n² + 2n + 1 - n² = 2n+1. Therefore, n can be any integer.

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More questions in this quiz

Answers and explanations are in the full quiz.

  1. What is the ones digit of the square of 12?
  2. The square of an even number is always:
  3. Find the number of trailing zeros in the square of 100?
  4. Prove that the square of any odd number is of the form 8k+1, where k is an integer.
  5. The square of an odd number is always of the form 8k+1 for some integer k. Prove or disprove this statement.

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