Common factors: which jump sizes can land on a number

These teaching notes are for members

What the video covers, what to say before it, where a class usually goes wrong, and what to set afterwards. An account is free, and it opens every chapter of every book.

Create a free account Sign in

What to assume they know

Common multiples: why the first "idli-vada" is 15: multiples, common multiples, and reading a two-region overlap diagram
Division with remainder, and recognising a remainder of zero as "goes exactly"
Multiplication tables to 10, well enough to spot a factor pair
Reading a number line and marking equally spaced points on it
Skip-counting from 0 rather than from 1 — the jumps in this topic start at 0

What they should be able to do

List every factor of a number under 100 systematically, in pairs
State that a jump size lands on a number exactly when it is a factor of it
Use both of the book's words, factor and divisor, for the same idea
Find the common factors of a pair by intersecting the two factor lists
Explain why 1 is a common factor of every pair, and why no common factor can exceed the smaller number
Contrast the finite list of common factors with the endless list of common multiples, and say why the difference is forced
Read a shaded-and-circled number grid as two multiple-lists laid over each other
Work backwards from clues about factors and digits to identify a number

Where it usually goes wrong

"1 and the number itself do not really count as factors." The book has to add them back explicitly on p.109 because they are the two everyone drops. Every factor list starts at 1 and ends at the number, always.
"Common factors of a big pair should be a long list." Size has nothing to do with it. 14 and 36 are not small, and they share exactly two factors. What controls the length is what the pair holds in common, not how large it is.
"Common factors and common multiples behave the same way." They do not, and this is the topic's second argument. Factors of a number are trapped between 1 and the number, so any pair has finitely many common factors and always at least one. Multiples run off to the right for ever.
"A hop of 7 reaches 36 because 36 is bigger than 7." Landing is about dividing exactly, not about getting far enough. Walk the arcs past 36 and let the miss be visible.
"Shaded means 'multiple of the smaller number' automatically." The grid on p.110 is deliberately unlabelled. The class has to find which rhythm is shaded and which is ringed; handing them the answer removes the whole task.
"A perfect number is one that looks neat." It is defined by a computation, and the definition is printed inside the question. Add the factors, compare with twice the number, and there is no aesthetic judgement anywhere in it.

Questions to check understanding

List all the factors of a given two-digit number
Say whether a stated hop length will land on a stated number, and justify it
Find the common factors of a given pair, or of three given numbers
Give the hop lengths that reach both of two treasures
Explain why every pair of numbers has at least one common factor
Identify a number from clues about its factors and its digits
Decide whether a number is perfect by adding its factors
The appendix bound with this chapter answers the §5.1 questions on footer pages 1 to 3

Examples worth working on the board

The hopping game (§5.1, p.109). One player hides a treasure on a number. The other picks a hop length and must keep it, starting from 0 and landing only on multiples of that length. The treasure is won only if a hop lands exactly on it. The worked instance puts the treasure on 24 and tries a hop of 4.
The number-line figure (p.109). Checked against the printed page. A line marked 0 to 24 in unit ticks, with several sets of coloured dotted arcs springing over it — short arcs above the line for the smaller hops, long arcs below it for the larger — each set landing on 24. Redraw it; the point of the drawing is that several different arc-lengths converge on the same endpoint.
The factor list of 24 (p.109). As printed: 1, 2, 3, 4, 6, 8, 12, 24. The book reaches it in two goes — a first sweep that finds 2, 3, 4, 6, 8 and 12, then a separate remark restoring 1 and 24, which students routinely forget. Show it built as pairs that multiply to 24, so nothing can be dropped.
Two treasures, 14 and 36 (pp.109–110). A hop of 7 gives 7, 14, 21, 28, 35, 42 — it lands on 14 and steps straight over 36. Printed factor lists: for 14, 1, 2, 7, 14; for 36, 1, 2, 3, 4, 6, 9, 12, 18, 36. The overlap is 1 and 2. Note that the book's printed list for 36 stops at 36 and includes it.
A second pair, 15 and 30 (p.110). Asked in the margin, with a note that there is more than one answer.
The shaded-and-circled grid (p.110). Checked against the printed page. A four-row table holding 31 to 70, ten to a row. Green shading falls on 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66 and 69; blue rings fall on 32, 36, 40, 44, 48, 52, 56, 60, 64 and 68. Three cells carry both: 36, 48 and 60. The three questions printed under it ask what the shaded cells share, what the ringed cells share, and what the doubly marked cells are called.
The closing Figure it Out (p.111). Useful inputs: multiples of 40 between 310 and 410; a number below 40 with 7 among its factors and digits summing to 8; a number below 100 with both 3 and 5 among its factors whose two digits differ by 1; the common factors of 20 and 28, of 35 and 50, of 4, 8 and 12, and of 5, 15 and 25; three multiples of 25 that are not multiples of 50; and hop lengths that reach both 28 and 70.
Perfect numbers (p.111, question 3). The book defines the term in the question itself and works 28 as the example: its factors 1, 2, 4, 7, 14 and 28 total 56, which is twice 28. The reader is then asked for one between 1 and 10.

Figures to have open

The p.109 number line with hop arcs. Must be redrawn, but the redraw has to keep the book's essential feature: several different arc lengths all reaching the same number.
The 31-to-70 grid with two independent markings. The markings must match the printed ones exactly, since the student is looking at the same grid.
A factor-pair ladder for a chosen number, showing pairs closing in from both ends. Standard schematic; the book does not print one, and it is the cleanest way to show that a factor list cannot be incomplete.
A two-list overlap diagram, reused from the previous topic so the student sees the same picture serving factors that served multiples.
No photograph or textbook data table is needed.

Where this sits in the book

NCERT Ganita Prakash, Class 6, Chapter 5 "Prime Time", §5.1 "Common Multiples and Common Factors", pp.109–111 — the hopping game and the number line (p.109), the two-treasure case and the marked grid (p.110), and the closing Figure it Out (p.111)
Chapter Summary, p.128, for the book's printed definition of factor
Forward pointer inside the same chapter: §5.4, pp.120–122, where the same question is answered from prime factorisations instead of full factor lists
Solutions appendix bound with this chapter file, footer pages 1–3, for §5.1