5. Number Play

Part I, printed Chapter 511 topics2 h

Chapter 5 of NCERT Mathematics for Class 8 (Part I, printed Chapter 5): Number Play. Three sections — Divisibility as something you can argue about, Divisibility shortcuts, and why they work and Digits in disguise. 11 videos, 2 h in all.

Worked answers to the 32 exercise questions in this chapter →

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Part 1

Divisibility as something you can argue about

10 minWhen a sum of consecutive numbers is a multiple of somethingA run of consecutive numbers is not a loose pile. Its total is pinned to its middle — and whether there is a middle decides everything.10 min · 12 partsहिंदी10 minDeciding parity without computing the answerWhether a total is even or odd is fixed before you compute it. Write 3, 4, 5, 6 and count the gaps rather than the numbers.10 min · 12 partsहिंदी11 minThe four divisibility facts you can prove, and how to use themBeing a multiple of something means being built out of complete rows. Once that is the picture, most of the divisibility facts stop needing proof.11 min · 12 partsहिंदी10 minAlways, sometimes, or never: one counterexample settles itAlways, sometimes and never are not three levels of confidence. They are three shapes of claim, and each is earned a different way.10 min · 12 partsहिंदी10 minWriting "leaves remainder r" as an algebraic expression“Leaves 3 when you divide by 5” is not a fact about a number. It names an endless, evenly spaced family.10 min · 12 partsहिंदी

Part 2

Divisibility shortcuts, and why they work

10 minThe digit-sum test for 9, and the algebra underneath itAdding up digits looks like it has nothing to do with dividing. It works entirely because of what our place values leave behind.10 min · 12 partsहिंदी10 minWhy the same digit sum also settles divisibility by 3The digit test for three is not a second rule. It is the test for nine, reused without a single new idea.10 min · 12 partsहिंदी9 minThe alternating-sum test for 11The test for eleven looks nothing like the test for nine, and is built by the same method with exactly one line changed.9 min · 12 partsहिंदी10 minDivisibility by 6 and other numbers, checked through their factorsSix has no divisibility test of its own and does not need one. Test 2, test 3, and a number passing both is a multiple of 6.10 min · 12 partsहिंदी10 minDigital roots, and what survives repeated digit-summingSumming a number's digits over and over destroys almost everything about it. Exactly one thing survives.10 min · 12 partsहिंदी

Part 3

Digits in disguise

10 minCracking a cryptarithm by reasoning about digits, not guessingA sum with letters where the digits should be looks like a search. It is not — it is a chain of local facts, each one forced.10 min · 12 partsहिंदी

What you will be able to do

  • Write a chosen number as a sum of two or more consecutive numbers, and say when this can be done in more than one way
  • List all eight ways of placing plus and minus signs between four numbers, using a branching diagram rather than trial and error
  • Sort even numbers into two classes by what they leave on division by 4, and write each class with a letter-number
  • Classify a divisibility statement as always, sometimes or never true, and name the evidence the verdict rests on
  • Generate several numbers satisfying a stated remainder condition, and recognise the regular step between them
  • Write a general multi-digit number as a sum of its place values using letter-numbers, and identify each digit's contribution
  • Identify the exact step of the nines argument where the number 9 was used, and check whether 3 can replace it there
  • Express each place value as a multiple of eleven with an offset of one, and identify the sign of that offset
  • Test divisibility by 6 by testing 2 and 3, and verify the result by dividing
  • Compute the digital root of a number by summing digits repeatedly, and say when to stop
  • State the three rules a cryptarithm obeys and explain what each one rules out

What this chapter assumes you already know

  • Adding and subtracting whole numbers and integers, negatives included
  • Standing a letter in for an unknown whole number, and collecting like terms — the chapter calls such a letter a letter-number
  • Even and odd numbers, and reading "even" as "has 2 as a factor"
  • Division leaving a remainder
  • Even and odd numbers, including for negative integers
  • Letter-numbers standing for any integer, and simplifying an expression by collecting like terms
  • Taking a common factor out of a sum, and reading the result as "2 times something"
  • Squaring a number, and the meaning of a cube written as a small raised 3

It builds directly on What makes a number a perfect square and Powers of 10, and place value written out for whole numbers and decimals.

Where people usually slip up

Sentences students actually say, taken from the notes the videos were made from. Each one is answered on the page of the video it belongs to.

  • "Consecutive means any increasing list."
  • "You test it on a few runs and then you know."
  • "Odd-length runs are multiples of their length, so even-length runs are too."
  • "An even number is a sum of consecutive numbers, so it is a sum of two."
  • "The sum of four consecutive numbers is a multiple of 4 because there are four of them."
  • "Negative numbers cannot be part of a run."
  • "Negative numbers are neither even nor odd."
  • "Checking a few cases shows it always happens."