The four divisibility facts you can prove, and how to use them

What to assume they know

• Factors, multiples, and reading "divisible by 8" as "8 times a whole number"
• Prime factorisation of a whole number, and LCM
• Letter-numbers, and taking a common factor out of a sum or difference
• Removing brackets and multiplying out, including with a minus in front
• Remainders on division, and the fact that a remainder is smaller than the divisor
• Even and odd numbers as the case a = 2 of all of this (Deciding parity without computing the answer)

What they should be able to do

• Sort even numbers into two classes by what they leave on division by 4, and write each class with a letter-number
• Predict, before adding, whether the sum of two given even numbers is a multiple of 4, and justify the prediction with algebra and with a row diagram
• State and prove that a common divisor of two numbers divides their sum and their difference
• State and prove that every multiple of a multiple of k is itself a multiple of k
• State and prove that a number divisible by k is divisible by every factor of k
• Explain why two divisors together force divisibility by their LCM, and why the product is the wrong answer in general
• Choose which of the four facts a given question needs, and apply it
• Read a row diagram as a proof rather than as an illustration

Where it usually goes wrong

• "Two even numbers add to a multiple of 4." They do only when both leave the same amount on division by 4. The chapter's three-case table exists because this is the single most common wrong prediction in the section.
• "The row picture is a nice illustration; the algebra is the proof." They are the same proof told twice. Combining two row-blocks is factoring out the common divisor. Say this explicitly — Part I p.118 says the generalising is being done by both together.
• "If a divides the sum, it divides each part." This is statement 2, and it is only sometimes true: 72 splits as 48 + 24, where 8 divides both, and as 50 + 22, where it divides neither. The implication runs one way only.
• "Divisible by 7 means divisible by 14, 21, 28 and the rest." Statement 5. Divisors travel downward to factors, never upward to multiples. 42 is the chapter's own witness.
• "Divisible by 6 and by 4 means divisible by 24." The two divisors share a factor of 2, so it gets counted twice. The right answer is the LCM. The chapter's own counterexample, 12, sits on Part I p.130.
• "So the rule is always the LCM, and never the product." The product is right precisely when the two divisors share no prime — which is why 9 and 4 do force 36. Give the student the condition, not just the warning.
• "A remainder can be anything." On division by 4 an even number leaves 0 or 2 and nothing else. The two-class split at the start of the section is a complete classification, and the completeness is what makes three cases enough.

Questions to check understanding

• Given two even numbers, say whether their sum is a multiple of 4 before adding, and justify it
• Complete the missing case of a three-case divisibility table, supplying both the algebra and worked examples
• Prove, with letter-numbers, that a common divisor of two numbers divides their difference
• Decide whether a divisibility claim is a correct use of one of the four facts, and name the fact
• Given that a number is divisible by two stated numbers, state the largest divisor you can be sure of
• Choose the Venn diagram that correctly nests multiples of 4, 8 and 32, and say why the others fail
• Explain, using prime factorisation, why one pair of divisors can be multiplied together and another pair cannot

Examples worth working on the board

Inputs. Values marked "printed" are the chapter's own; the rest are its questions.

• The two classes of even number (Part I, §5.1, p.116, two figures side by side). Left: a block of blue dots in complete rows of four, captioned as the even numbers that are multiples of 4 and leave nothing over. Right: the same kind of block with a short final row of two dots, captioned as the even numbers that leave 2. Both blocks have a vertical dotted ellipsis to show they continue. Read off the images, not the text layer — the caption sits under artwork.
• Case one, both multiples of 4 (Part I, §5.1, p.117, first row of the table). Algebra printed: 4p and 4q, then 4p + 4q = 4(p + q). Picture: a p-row block of blue fours plus a q-row block of yellow fours, equalling a single (p + q)-row block. Examples printed: the list 4, 12, 16, 24, 36; then 12 + 16 written as 4(3 + 4) = 28; and 16 + 28 as 4(4 + 7) = 44.
• Case two, neither a multiple of 4 (Part I, §5.1, p.117, second row). Algebra printed: (4p + 2) and (4q + 2), then their sum as 4p + 4q + 4 = 4(p + q + 1). Picture: two blocks each with a short row of two, combining into a block of (p + q + 1) rows — the two short rows fusing into one complete row, which the figure labels. Examples printed: the list 2, 6, 10, 18, 22, 42; then 2 + 6 = 8, 6 + 10 = 16, 22 + 6 = 28.
• Case three, one of each (Part I, §5.1, p.118, top of page). Algebra printed: 4p + (4q + 2) reduced to 4(p + q) + 2. Picture: the combined block has (p + q) complete rows and a leftover 2, which the figure labels. The Examples cell for this case is blank on the page and the explanation cell is blank too — the chapter hands the whole case to the student.
• Statement 1 and its table (Part I, §5.1, p.118). Claim: a number dividing two numbers separately divides their sum, put for the divisor 8. Algebra printed: 8a and 8b, then 8a + 8b = 8(a + b). Picture: an a-row block of blue eights above a b-row block of yellow eights, bracketed as (a + b) rows. Examples printed: the pairs 8 and 16, 16 and 56, 80 and 120; then 8 + 16 written as 8(1 + 2) = 24, 16 + 56 = 72, 80 + 120 = 200. The page then asks whether the same holds for subtraction — that half is the student's, and the general statement on Part I p.119 covers both.
• Statement 2 (Part I, §5.1, p.119). Claim: if a number is divisible by 8, then 8 divides any two numbers that add up to it. Algebra printed: 8m split as 8a + 8b, and also as p + q with p and q not multiples of 8. Examples printed: the list 8, 16, 56, 72; then 72 = 48 + 24, annotated as 8 × 9 = 8 × 6 + 8 × 3; and 72 = 50 + 22. Verdict printed: sometimes true.
• Statement 3 (Part I, §5.1, p.119). Claim: every multiple of a number divisible by 7 is divisible by 7. Algebra printed: 7j, then (7j) × m, with the row count mj. Examples printed: 14 = 7 × 2 with j = 2, 42 = 7 × 6 with j = 6, 98 = 7 × 14 with j = 14; then multiples of 14: 28 = (7 × 2) × 2, 70 = (7 × 2) × 5, 154 = (7 × 2) × 11. Verdict printed: always true, with the general form given as: if A is divisible by k, then every multiple of A is divisible by k.
• Statement 4 (Part I, §5.1, p.120). Claim: a number divisible by 12 is divisible by every factor of 12. Algebra printed: 12m, rewritten as 2 × 6 × m and as 3 × 4 × m. Picture: an m-row block of twelves with a caption saying that a factor of 12 covers a row fully and therefore covers the whole block. Examples printed: the list 12, 24, 36, 48, 108, 132; and the factors of 24 listed as 1, 2, 3, 4, 6, 8, 12, 24. Verdict printed: always true, generalised as: if A is divisible by k, then A is divisible by every factor of k.
• Statement 5 (Part I, §5.1, p.120). Claim: a number divisible by 7 is divisible by every multiple of 7. Algebra printed: 7k and 7m, with the condition that 7k is divisible by 7m exactly when m is a factor of k, and the division worked as 7ym ÷ 7m = y when k = ym. Examples printed: 42, written as 7 × 6, is divisible by 7 but not by 28, written as 7 × 4; and 42 is divisible by 14, written as 7 × 2. Verdict printed: sometimes true.
• Statements 6 and 7, and the LCM rule (Part I, §5.1, p.121). Statement 6: divisible by both 9 and 4 forces divisibility by 36. Statement 7: divisible by both 6 and 4 forces divisibility by 24. Both are flagged "Math Talk" and left to the class; the general rule printed immediately after is that two divisors force divisibility by their LCM, justified by prime factorisation. Note: the counterexample that settles statement 7 is printed later in the same chapter — Part I p.130 points out that 12 is a multiple of 4, and also of 6, and yet not of 24. Use the chapter's own number rather than inventing one.
• The SUMMARY card (Part I p.134). Four bullets, in this order: every multiple of a multiple; every factor of a divisor; sums and differences; the LCM rule. The explanation's section 12 should mirror this order so the student's revision card and the explanation agree.
• Venn diagram item (chapter-end "Figure it Out", Part I p.133 no. 16, with the four options drawn on Part I p.134). Four candidate pictures for the relationship between multiples of 4, multiples of 8 and multiples of 32: (i) three circles in a row, each overlapping only its neighbour; (ii) two overlapping circles with the middle label pointing at the overlap; (iii) three nested circles with 32 named on the outermost ring and 4 on the innermost; (iv) three nested circles with 4 on the outermost ring and 32 innermost. Hand the four options over as data; the choice is a direct test of facts two and three.

Figures to have open

• The two-class dot figure for even numbers under division by 4 (Part I, §5.1, p.116). The chapter's own figure and the foundation of the whole section; redraw as a schematic.
• The three-case combining diagram (Part I, §5.1, pp.117–118). The chapter draws cases one and two and leaves case three's example cell empty; the explanation needs all three drawn to the same convention, with the leftovers colour-coded.
• A generic row block of a rows of k, reusable for statements 1 to 5. Standard schematic; the chapter uses this same object on every page from Part I p.117 to p.120.
• Prime-factor bars for pairs of divisors — 9 and 4, then 6 and 4 — with the shared prime visibly double-counted in the second. Standard schematic. The chapter argues this in words on Part I p.121 and draws nothing.
• The four Venn options from Part I p.134, redrawn.
• No photograph is needed.

Where this sits in the book

• NCERT Ganita Prakash Class 8, Part I, printed Chapter 5, "Number Play", §5.1 "Is This a Multiple Of?", printed subheadings "Pairs to Make Fours" (Part I pp.116–118) and "Always, Sometimes, or Never" (Part I pp.118–121).
• The eight numbered statements and their verdicts are printed at Part I pp.118–120 (statements 1–5) and Part I p.121 (statements 6–8), with no verdict printed for 6 and 7; the tables carry the heading "Explanation with Algebra and Visualisation".
• The general forms are printed as running bold lines at Part I p.119 (sums and differences), Part I p.120 (multiples, and factors) and Part I p.121 (the LCM rule).
• The chapter's SUMMARY, Part I p.134, restates all four.
• Chapter-end "Figure it Out", Part I p.133 nos. 12 and 16, with the Venn options drawn on Part I p.134.
• Forward pointer inside the same chapter: the counterexample for statement 7 appears at Part I p.130, in the discussion of testing divisibility by 24.