Divisibility tests for 10, 5, 2, 4 and 8: why only the last digits matter

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 factors: which jump sizes can land on a number: factors, and the fact that "is a factor of" and "divides exactly" are one idea
Place value in numbers up to five digits — that a numeral is built out of ones, tens, hundreds and thousands
Common multiples: why the first "idli-vada" is 15: multiples, and continuing a run of them
Long division, well enough to know what the shortcuts are saving

What they should be able to do

Test a number for divisibility by 10, 5 and 2 from its last digit
Test for divisibility by 4 from the last two digits, and by 8 from the last three
Show, with a counterexample, why the last digit alone cannot settle divisibility by 4
Explain, in terms of tens, hundreds and thousands, why the tail is enough
Say why the same style of test does not exist for 3, 6, 7 and 9
Use one test to obtain several others — recognise that some checks make others unnecessary
Apply the tests to leap years, palindromes and factor puzzles
State a divisibility claim in both directions and judge whether each direction holds

Where it usually goes wrong

"If it works for 2, 5 and 10, the last digit works for everything." The four printed counterexample pairs on p.124 exist to break this, and the break should be felt before the fix is offered.
"A number ending in 4 is divisible by 4." 14, 34 and 54 are not. Ending in a multiple of 4 and being a multiple of 4 are different claims.
"The rule for 4 is 'the last two digits are divisible by 4', which is just another rule to memorise." It is a consequence of a hundred being a multiple of 4. Section 8 is where it stops being arbitrary, and the same sentence then gives away section 9 for free.
"A number divisible by 4 must be divisible by 8." 12, 20 and 28 are not. The windows widen; the divisibility does not automatically follow.
"A statement and its converse are the same claim." The book prints both directions for 4 and again for 8 and asks about each. In these two cases both directions do hold — which is precisely why a student who never separates them gets no warning that they can come apart.
"To show a number is divisible by 2, 4, 5, 8 and 10 you must run five tests." Question 5 on p.126 exists to deny it. Some of these divisors carry others, and finding which is the real question.
"Any four-digit palindrome ending in an even digit is divisible by 4." The first digit and last digit of a palindrome are the same, so this constrains the number far more than it looks. Working it out is question 2, and it is a better puzzle than it first appears.

Questions to check understanding

State whether a number you are handed is a multiple of 2, 4, 5, 8 or 10, and say which digits you used
Give a counterexample to the claim that the last digit settles divisibility by 4
Find the multiples of 4, or of 8, inside a stated narrow range
Change the tail of a given number to make it a multiple of a stated divisor
Say which two of a list of divisibility checks make the rest unnecessary
Find the remainders of several numbers on division by 10, by 5 and by 2
Count the leap years in a stated span
Write two numbers with a stated product, neither ending in zero
The appendix bound with this chapter answers the §5.5 questions on footer pages 9, 10 and 11 — with one answer that is wrong, see Notes

Examples worth working on the board

The opening problem (§5.5, pp.122–123). The number 8560 is put up and the question is which of 2, 3, 4, 5, 6, 7, 8, 9 and 10 are factors of it. The section's promise is that several of these can be settled without dividing. 8560 returns as a test case in every subsection, so keep it in view throughout.
Divisibility by 10 (p.123). The run 10, 20, 30, 40, … is continued and its ending inspected; 125 is offered as a number to place in or out of that run. The claim to be judged is that the multiples of ten are exactly the numerals ending in zero.
Divisibility by 5 (p.123). The run 5, 10, 15, 20, 25, … The largest number below 399 divisible by 5 is asked for, and 8560 is tested again. The claim to be judged names two possible endings.
Divisibility by 2 (pp.123–124). The run 2, 4, 6, 8, 10, … up to 20. Then 682 and 8560 as test cases, and the multiples of 2 between 399 and 411 as a short listing exercise. The claim names five possible endings.
Why the last digit fails for 4 (p.124). Four printed counterexample pairs: 12 against 22, 14 against 24, 16 against 26, and 18 against 28. In each pair the two numbers end alike and disagree about 4. This is the pivot of the whole section and it is worth working through all four pairs, not one.
Finding the pattern for 4 (p.124). The reader is asked to list the multiples of 4 up to 200 and hunt for structure, then to find the multiples of 4 between 330 and 340, between 1730 and 1740, and between 2030 and 2040. The three windows are chosen so that the surviving two-digit tails are the same in all three, which is the observation the test rests on.
The test for 4, as three separate claims (p.124). That nothing above the final pair of digits can affect the verdict; that a two-digit tail which is a multiple of 4 forces the whole number to be one; and the converse, that a whole number which is a multiple of 4 forces its tail to be one. The book prints all three and asks whether the reader agrees with each. Treat them as three claims, because separating a statement from its converse is a skill this section is quietly teaching.
Divisibility by 8 (p.125). Multiples of 8 between 120 and 140, between 1120 and 1140, and between 3120 and 3140 — the same three-window construction, now needing three digits. Then the same three claims, restated for three digits.
The 8560 instruction (p.125). The reader is asked to alter the final pair of digits in 8560 until what stands there is a multiple of 8. See Notes: 8560 already is one, so this must be read as asking for a different multiple of 8 beginning 85.
Where the shortcuts stop (p.125). The section closes by saying that simple methods have been found for 10, 5, 2, 4 and 8, and that tests for 3, 6, 7 and 9 are left to later classes. Checked against the printed page.
The cheapest sufficient pair (p.126, question 5). A teacher asks whether 14560 passes all five of 2, 4, 5, 8 and 10; a child checks only two of them and declares the rest. More than one pair works.
Bulk testing (p.126, question 6). Which of 572, 2352, 5600, 6000 and 77622160 pass all five tests. Excellent for showing the tests running side by side on a table.
Leap years (p.125, question 1). The rule as printed: a year qualifies when 4 divides it, unless 100 divides it and 400 does not. The reader is asked which years since their birth were leap years, and how many leap years fall from 2024 to 2099 inclusive. Note that the exception clause does not bite anywhere in that second range, which is worth saying rather than leaving as a silent simplification.
Palindromes (p.125, question 2). The largest and smallest four-digit palindromes divisible by 4. See Notes — the appendix's answers to this one are wrong.
Two factors of 10000 with no trailing zero (p.126, question 7). A neat bridge back to §5.4: split the prime factorisation of 10000 so that no piece gets both a 2 and a 5, and neither piece can end in zero.

Figures to have open

A numeral with a sliding window over its tail, widening from one digit to two to three. This is the topic's signature figure; the book prints nothing like it, and it carries sections 6, 7 and 8 at once.
A split-numeral diagram for section 8: the number written as a left-hand part that is visibly a whole number of tens, hundreds or thousands, plus the tail. Standard schematic and the only place the argument actually lives.
A results table for the bulk test in question 6, five divisors across and five numbers down. Standard schematic.
A number axis able to carry a run of multiples with their final digits picked out, reused across sections 2, 3 and 4.
No photograph or textbook data table is needed. §5.5 prints no diagram — checked against pp.122 to 126.

Where this sits in the book

NCERT Ganita Prakash, Class 6, Chapter 5 "Prime Time", §5.5 "Divisibility Tests", pp.122–126 — the opening problem (pp.122–123), the tests for 10, 5 and 2 (p.123), the failure of the last digit and the test for 4 (p.124), the test for 8 and the deferral of 3, 6, 7 and 9 (p.125), and the Figure it Out (pp.125–126)
§5.6 "Fun with Numbers", pp.126–128, follows this section and is indexed in the spine as a non-video topic; nothing from it is needed here
Backward pointer inside the same chapter: §5.4, pp.120–122, for the factorisation view of the same question
Solutions appendix bound with this chapter file, footer pages 9–11, for §5.5