Toggles equal factors
A hundred lockers, and one of them tells you everything
Person 3 touches lockers 3, 6, 9 and so on, so a locker is touched once by each person whose number divides it.
📚 Chapter Learning Kit
A Square and a Cube · Class 8 Mathematics · Ganita Prakash
English 3 topics 105 leveled MCQs 51 flashcards 5 games Free
Shared by a Veda teacher · Generated with Veda AI · Oct 2026
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✏️ Example
The whole idea, one bite at a time
Veda Bites are swipeable micro-lessons — each one teaches exactly one idea. Here's a taste from this kit; the app has the full deck.
Factors travel in pairs
Why most lockers end up shut
Every factor has a partner that multiplies with it to give the number. For 6 the pairs are 1 with 6 and 2 with 3.
A factor can partner itself
The one case where a pair is a single
For 36 the pair 6 with 6 has both members the same, so it adds one factor and not two.
A root undoes a square
The side of a square, from its area
A square tile of area 49 square units has a side of 7 units, because 7 times 7 is 49.
Positive and negative
Sixty-four has two roots, not one
Eight times eight is 64, and minus 8 times minus 8 is also 64.
Listing until you arrive
Counting up to a square
To test 576, write the squares from 20 squared: 400, 441, 484, 529, 576. It appears, so 576 is 24 squared.
📖 Smart notes
What you'll study, topic by topic
1
Squares and the patterns they keep
A square is a number times itself. Squares are the only numbers with an odd count of factors, which is why the square-numbered lockers stay open. Their last digits, their trailing zeros and their gaps all follow rules wo…
- A square is a number multiplied by itself, written n squared.
- Factors come in partner pairs, so most numbers have an even count.
- A square has one pair whose two factors are equal, so its count is odd.
~35 min · full explanation, examples & memory tricks in the app
2
Square roots and testing a perfect square
A square root undoes a square. Every perfect square has two integer roots, one positive and one negative. Prime factorisation settles whether a number is a square, and nearby squares give a quick estimate of a root.
- If y is x times x, then x is a square root of y.
- A square of area 49 square units has a side of 7 units.
- A perfect square always has two whole-number roots, one of each sign.
~35 min · full explanation, examples & memory tricks in the app
3
Cubes, cube roots and where the words came from
A cube is a number multiplied by itself three times. Its prime factors split into three identical groups, which both tests a cube and names its root. The words square and root carry their own history.
- A cube is a number multiplied by itself three times, written n cubed.
- A cube of edge 4 units holds 64 unit cubes, in four layers of sixteen.
- Cubes can end in any digit, unlike squares.
~35 min · full explanation, examples & memory tricks in the app
❓ Leveled MCQ practice
Try the smart MCQs from this kit
105 questions laddered from warm-up to topper-level, each with an explanation. A taste:
A locker is toggled once by every person whose number divides the locker number, and it starts closed. It ends up open when the count of those people is
Intermediate
A even
B odd
C prime
D a square
Show answer & explanation
odd
Each toggle flips the state. Starting closed, an odd number of flips leaves it open and an even number leaves it closed.
Locker 6 is touched by persons 1, 2, 3 and 6 and by nobody else, so the count of people who touch it is the count of its
Beginner
A multiples
B digits
C prime factors
D factors
Show answer & explanation
factors
Person k touches the locker exactly when k divides the locker number, so the people who touch it are its factors.
Factors are listed in partner pairs that multiply to the number, which is why the usual count of factors comes out
Intermediate
A odd
B prime
C even
D a square
Show answer & explanation
even
Pairs supply two factors at a time, so the total is even unless some pair has both members equal.
For the number 36 the factor pairs run 1 with 36, 2 with 18, 3 with 12, 4 with 9 and then 6 with 6. That last pair is unusual because it
Advanced
A adds three factors rather than two
B adds only one factor instead of two
C repeats a pair already counted
D uses a factor that is not whole
Show answer & explanation
adds only one factor instead of two
Both members are 6, so the pair contributes a single factor. That is what makes the total count odd.
🃏 Flashcards
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51 flashcards in this kit — the app reviews them with spaced repetition so the right card returns on the right day.
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Memory Match
Word Match
Categorization
Fact Or Myth
Sequence Builder
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