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CBSE Class 8 · Mathematics

Number Play

Official NCERT chapter from Ganita Prakash Part I (book code hegp1). ExamMaster notes are original teaching at CBSE Class 8 depth.

This lesson follows the official chapter “Number Play” in Ganita Prakash Part I. The words below are ExamMaster’s teaching, not a paste from the book. Use the NCERT chapter for the classroom sequence; use these notes to hold the idea without copying exercises or figures.

  • CBSE Class 8
  • Easy level
  • 8 concepts

1Is This a Multiple Of?

“Is this a multiple of?” means: does a whole k exist with n = k × d? 84 is a multiple of 7 because 12 × 7 = 84. 85 is not.

Ending in 4 is not a multiple-of-7 test.

Figure. A multiple of 6 is a number that lands on an equal-group tick: 0, 6, 12, 18, 24, 30. 18 is 6\times 3, remainder 0. 20 sits between 18 and 24, so it is not a multiple of 6. The test is the landing, not a tidy-looking gap.

How it works

  1. Name n and d84 and 7.
  2. Divide or factor12 × 7 = 84.
  3. Yes if no remainder; no if a remainder staysThe play.

2Checking Divisibility Quickly

Quick checks use last digits or digit-sums as taught: even last digit for 2; 0 or 5 for 5; digit-sum for 3 or 9; last two digits for 4, as the lesson framed them. A quick check is a filter; a hard n still needs a divide if the filter is silent.

A last-digit rule for 7 that you invent today is not this chapter’s quick check.

Figure. Quick tests read digits, not long division. Last digit decides 2, 5 and 10 (here 4 is even, so 414 is a multiple of 2). Digit sum decides 3 and 9: 4+1+4=9, and 9 is itself a multiple of 9, so 414 is a multiple of 3 and of 9.

How it works

  1. Pick the d you were taught a shortcut for2, 3, 5, 9, 4…
  2. Run that shortcutLast digit or digit-sum.
  3. If it fails, n is not a multiple; if it passes, you may still confirm by dividingA filter.

Digit-sum for 3

Is 258 a multiple of 3?

  • Digit-sum2+5+8=15
  • 15 ÷ 35
  • Yes258 = 86 × 3

Pro tip. Digit-sum divisible by 3 means the number is.

3Digits in Disguise

A digit in disguise is a letter standing for a digit 0–9 in a cryptarithm, or a hidden house. Unmask means name the digit and check the sum still holds.

A pretty letter is not a digit until you say which 0–9 fits.

Figure. Digits in disguise: a letter stands for one digit 0–9, and the same letter means the same digit in every house. A4A is 100A+40+A=101A+40, not “A then four then A as words”. The tens house is already known (4); only A is hidden.

How it works

  1. See the disguiseA letter in a sum.
  2. Try a digit that fits the columnsThe unmask.
  3. Check every columnThe disguise worked.

4A definition is a test you can run

Multiple-of is a test: remainder 0 when you divide by d. If you cannot run divide or a taught shortcut, you have a guess.

“It looks even” for d = 3 is the wrong filter.

Figure. “Multiple of 6” is a test you can run: divide and read the remainder. 24 is 6\times 4 with remainder 0, so it is a multiple. 25 is 6\times 4 with remainder 1, so it is not. A tidy drawing of five boxes is not the test.

How it works

  1. Name dThe divisor.
  2. Divide or run the taught shortcutThe test.
  3. Remainder 0 or failThe definition.

5Name the given before the unknown

The given are n and d (or the cryptarithm). The unknown is yes/no or the hidden digit. Copy n before you peek at a last digit that belongs to a different d.

Using the 5-rule on a 3-question is a silent swap.

Figure. Name the given before the unknown. In 2A4 the 2 and the 4 are already ink; A is the ask. For divisibility by 3, the digit sum 2+A+4=6+A must be a multiple of 3, so A can be 0, 3, 6 or 9. Do not chase A before writing those two givens.

How it works

  1. Copy n and dThe given.
  2. Name yes/no or the letterThe unknown.
  3. Then the matching shortcut or divideGiven first.

6One worked case is enough at this class

One divisibility check is enough. A page of “is 84 a multiple of 7” clones does not add a new filter.

Ten twins of 12×7 are still one idea.

Figure. One worked case at this class: is 414 a multiple of 9? Sum the digits, 4+1+4=9, and 9 is 9\times 1. The test passes. You do not need a second number to see how the digit-sum rule runs.

How it works

  1. Run one checkThe case.
  2. A second only as a check of a different dOptional.
  3. StopThis class.

7A miss: swapping the school name for the picture

A locker-code story is a setting. The maths is n = k × d. Spinning a story-dial and skipping the remainder is the swap-miss.

A movie code is not 84 ÷ 7.

Figure. A miss: swapping the school name for the picture. Three boxes that look like “groups” but hold 5, 5 and 3 add to 13 — not a multiple of 6. Equal groups of 6, 6 and 6 add to 18 and pass the test. The word multiple is the remainder-zero test, not a tidy row of ink.

How it works

  1. Keep the locker as a storyA setting.
  2. Write n and dThe maths.
  3. DivideDo not swap.

8Check by the opposite action or the opposite test

Check a yes-claim by multiplying back: k × d must be n. Check a no-claim by showing a remainder. A check that is “the shortcut felt right” is not the opposite test.

Repeating the last digit is not multiply-back.

Figure. Check by the opposite action. If you claim 64\div 8=8, multiply back: 8\times 8=64. The product must return the original count. A remainder claim is checked the same way: 25=6\times 4+1.

How it works

  1. State yes or noThe claim.
  2. Multiply back or show the remainderThe opposite test.
  3. Mend if it failsHonest.
258 is a multiple of 3 because
  1. 2+5+8=15 and 15 is a multiple of 3
  2. It ends in 8
  3. It is even

Digit-sum is the taught 3-filter.

Notes

  • The official chapter title is “Number Play”. Teach the school test for that title, not a contest shortcut.
  • If a step needs a later class, stop. The next official chapter will pick it up.

Formulas

  • n is a multiple of d ⇔ n = k × d
  • digit-sum test for 3 and 9 (as taught)

Recap

Hold these pegs from the official chapter “Number Play”. The wording is ExamMaster’s teaching, not a textbook recap.

Is This a Multiple Of?
“Is this a multiple of?” means: does a whole k exist with n = k × d? 84 is a multiple of 7 because 12 × 7 = 84.
Checking Divisibility Quickly
Quick checks use last digits or digit-sums as taught: even last digit for 2; 0 or 5 for 5; digit-sum for 3 or 9; last two digits for 4, as the lesson framed them.
Digits in Disguise
A digit in disguise is a letter standing for a digit 0–9 in a cryptarithm, or a hidden house.
A definition is a test you can run
Multiple-of is a test: remainder 0 when you divide by d.
Name the given before the unknown
The given are n and d (or the cryptarithm).
One worked case is enough at this class
One divisibility check is enough. A page of “is 84 a multiple of 7” clones does not add a new filter.

Practise Number Play

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  • 1 quick check with worked explanations
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