Interpretation of Primary Divisibility Rules
ANKIT PATODI ivisibility rules which help to quickly identify if a ClassRoom JWJTJCJMJUZgiven number SVMFTis XIJDI divisible IFMQ by UP 2, RVJDLMZ 3, 4, 5, JEFOUJGZ 6, 8 and JG B 10 HJWFOare taught OVNCFS in the JT EJWJTJCMF upper primary CZ classes. However, BOE theD conceptualBSF UBVHIU background JO UIF VQQFS or the QSJNBSZ proofs behind DMBTTFT the )PXFWFS rules are UIF%rarely DPODFQUVBM dwelt upon. CBDLHSPVOE One of the PS UIF reasons QSPPGT could CFIJOE be that UIF most SVMFT BSFof SBSFMZus, i.e., EXFMU the teachers, VQPO 0OF are PG not UIF aware SFBTPOT of the DPVME logic CF behind UIBU NPTU the PG VT rules. JF Even UIF UFBDIFST if we are, BSF we OPU may BXBSF believe PG thatUIF MPHJD the proofs CFIJOE for UIF these SVMFTrules are &WFO beyond JG XF theBSF scope XF NBZ of CFMJFWFunderstanding UIBU UIF of QSPPGT children GPS UIFTF as SVMFTthey involveBSF CFZPOE complex UIF TDPQF algebraic PG VOEFSTUBOEJOH expressions and PG DIJMESFO BT UIFZinterpretation. JOWPMWF DPNQMFY These BMHFCSBJD two reasons FYQSFTTJPOT together BOE distance children JOUFSQSFUBUJPOfrom learning the ɨFTF logic UXP behind SFBTPOT the UPHFUIFS rules. EJTUBODF DIJMESFO GSPN MFBSOJOH UIF MPHJD CFIJOE UIF SVMFT This challenged me to put the explanation behind these rules ɨJTin a simple DIBMMFOHFE manner. NF UP I sharedQVU UIF this FYQMBOBUJPO with a group CFIJOE of teachers UIFTF SVMFT JOand B TJNQMF saw that NBOOFS it helped * TIBSFE them further UIJT XJUI in B their HSPVQ classrooms. PG UFBDIFST I BOEused TBX a few UIBU basic JU IFMQFE rules to UIFN justify GVSUIFS my reasoning: JO UIFJS DMBTTSPPNT * VTFE B GFX CBTJD SVMFT UP KVTUJGZ NZ SFBTPOJOH RULE 1: If any number is divisible by another, all its 36-&multiples *Gwill BOZ also OVNCFS be divisible JT EJWJTJCMF by that CZ number. BOPUIFS For BMM JUT example, NVMUJQMFTif 10 is divisible XJMM BMTP by CF 2 thenEJWJTJCMF all multiples CZ UIBU OVNCFS of 10 viz. 'PS 20, FYBNQMF 30, JG100, JT1000 EJWJTJCMF and so CZ on UIFO will also BMM NVMUJQMFT be divisible PG by WJ[2. BOE TP PO XJMM BMTP CF EJWJTJCMF CZ Let us take an example to understand this rule. If we say that -FU10 is VT divisible UBLF BO FYBNQMFby 2, it means UP VOEFSTUBOE that when UIJT we SVMF divide *G XF10 TBZobjects UIBU into JT groups EJWJTJCMF of CZ 2 each, JU NFBOT no object UIBU is XIFO left out. XF EJWJEF The PCKFDUT JOUPrepresentation HSPVQT PG below FBDI (Figure OP PCKFDU 1) depicts JT MFGU PVU the ɨF same: SFQSFTFOUBUJPO CFMPX 'JHVSF EFQJDUT UIF TBNF
Figure 1 'JHVSF Keywords: Patterns, divisibility, place value, multiples, visualisation, generalisation
Azim Premji University At Right Angles, March 2021 23
1 Now,/PX /PX /PX suppose TVQQPTF TVQQPTF TVQQPTF we XF XF IBWFhaveXF IBWF IBWF PCKFDUT objects PCKFDUT PCKFDUT JO in JO BOZ JO anyBOZ BOZ NVMUJQMF NVMUJQMF multiple NVMUJQMF PG PG ofPG 10, XF XF DBOXF we DBO DBO EJWJEFcan EJWJEF EJWJEF divide UIF UIF UIF PCKFDUT thePCKFDUT PCKFDUT objects JOUP JOUP JOUP TNBMMFS TNBMMFS into TNBMMFS smaller HSPVQT HSPVQT HSPVQT PG groups PG PG JO JO JO of 2 in theUIFUIF sameUIF TBNF TBNF TBNF fashion GBTIJPO GBTIJPO GBTIJPO (Figure 'JHVSF 'JHVSF 'JHVSF 2). 4JNJMBS Similar 4JNJMBS 4JNJMBS SFMBUJPOTIJQT SFMBUJPOTIJQT relationships SFMBUJPOTIJQT DBO DBO DBOcan CF CF WJTVBMJ[FE CF be WJTVBMJ[FE WJTVBMJ[FE visualized GPS GPS GPSBOZ BOZ for BOZ OVNCFS anyOVNCFS OVNCFS number BOE BOE BOE JUT JUT andNVMUJQMFT JUT NVMUJQMFT NVMUJQMFT its multiples.
'JHVSF'JHVSF 'JHVSFFigure 2
36-&36-& *G *G UXP UXP PS PS NPSF NPSF OVNCFST OVNCFST BSF BSF BMM BMM EJWJTJCMF EJWJTJCMF CZ CZ UIF UIF TBNF TBNF OVNCFS OVNCFS UIFO UIFO UIFJS UIFJS TVN TVN XJMM XJMM BMTP BMTP CF CF RULE36-& 2 : If *G two UXP orPS NPSF more OVNCFST numbers BSF are BMM all EJWJTJCMF divisible CZ UIF by TBNFthe same OVNCFS number, UIFO UIFJS then TVN their XJMM sum BMTP will CF also be EJWJTJCMFEJWJTJCMFEJWJTJCMF CZ CZ UIBU CZ UIBU UIBU OVNCFS OVNCFS OVNCFS 'PS 'PS 'PS FYBNQMF FYBNQMF FYBNQMF JG JG JG BOE BOE BOE CPUI CPUI CPUI BSF BSF BSFJOEJWJEVBMMZ JOEJWJEVBMMZ JOEJWJEVBMMZ EJWJTJCMF EJWJTJCMF EJWJTJCMF CZ CZ CZ UIFJS UIFJS UIFJS TVN TVN TVN JF JF JF divisible by that number. For example, if 24 and 40 both are individually divisible by 4, their sum i.e., 24 XJMM XJMM XJMM BMTP BMTP BMTP CF CF EJWJTJCMF CF EJWJTJCMF EJWJTJCMF CZ CZ CZ ɨJT ɨJT ɨJT DBO DBO DBO CF CF FBTJMZ CF FBTJMZ FBTJMZ TIPXO TIPXO TIPXO VTJOH VTJOH VTJOH DPVOUFST DPVOUFST DPVOUFST 'JHVSF 'JHVSF 'JHVSF + 40 = 64 will also be divisible by 4. This can be easily shown using counters (Figure 3):
24 = 4 6 40 = 4 10 × ×
64 = 4 (6 + 10) ×
'JHVSF 'JHVSF'JHVSFFigure 3
ThisɨJTɨJTɨJT TJNQMFsimple TJNQMF TJNQMF SFQSFTFOUBUJPO SFQSFTFOUBUJPOrepresentation SFQSFTFOUBUJPO DPVME DPVME DPVME could CF CF HFOFSBMJ[FE CF HFOFSBMJ[FE HFOFSBMJ[FE be generalized GPS GPS GPSUIF UIF UIF DPNCJOBUJPO DPNCJOBUJPO DPNCJOBUJPO for the PG combination PG UISFF PG UISFF UISFF PS PS NPSF PS NPSF NPSF OVNCFSTof OVNCFST OVNCFST three 8F 8F 8F XJMMor XJMM XJMM CFmore CF CF numbers. Wewillbe VTJOH B GFX CBTJD FYQBOTJPOT PG OVNCFST JO PSEFS UP VOEFSTUBOE IPX UIF DPODFQU PG QMBDF WBMVF QMBZT BO usingVTJOHVTJOH a B few GFX B GFXbasic CBTJD CBTJD expansionsFYQBOTJPOT FYQBOTJPOT PG of OVNCFST PG numbers OVNCFST JO inPSEFS JO order PSEFS UP VOEFSTUBOE UPto VOEFSTUBOE understand IPX IPX UIFhow UIF DPODFQU the DPODFQU concept PG QMBDF PG QMBDF of WBMVF place WBMVF QMBZT value QMBZT BO plays BO an JNQPSUBOU SPMF JO VOEFSTUBOEJOH EJWJTJCJMJUZ SVMFT -FU VT UBLF BO FYBNQMF importantJNQPSUBOUJNQPSUBOU role SPMF SPMF in JO VOEFSTUBOEJOHunderstanding JO VOEFSTUBOEJOH EJWJTJCJMJUZ divisibility EJWJTJCJMJUZ SVMFT rules. SVMFT -FU -FU Let VT UBLF VT us UBLF take BO FYBNQMF BO an FYBNQMF example: 8F IBWF B OVNCFS XIJDI XF DBMM UIJSUFFO UIPVTBOE GPVS IVOESFE BOE ëGUZëWF 8F DBO FYQBOE We8F have8F IBWF IBWF a B number OVNCFS B OVNCFS 13455, XIJDI which XIJDI XF we DBMMXF call DBMM UIJSUFFO thirteen UIJSUFFO UIPVTBOE UIPVTBOE thousand, GPVS GPVS fourIVOESFE IVOESFE hundred BOE BOE ëGUZëWF and ëGUZëWF fifty-five. 8F 8F DBO DBO FYQBOE FYQBOE We can expand UIJTUIJT OVNCFS OVNCFS JO JO NBOZ NBOZ XBZT XBZT VTJOH VTJOH UIF UIF QMBDF QMBDF WBMVF WBMVF DPODFQU DPODFQU 4P 4P DBO DBO CF CF XSJUUFO XSJUUFO JO JO BOZ BOZ PG PG UIF UIF GPMMPXJOH GPMMPXJOH thisUIJT number OVNCFS in JO many NBOZ XBZT ways VTJOH using UIF the QMBDF place WBMVF value DPODFQU concept. 4P So, 13455 DBO CF can XSJUUFO be written JO BOZ PG in UIF any GPMMPXJOH of the following XBZTXBZTXBZT ways: t •t10000 t + 3000 + 400 + 50 + 5 •t13000t t + 400 + 50 + 5 = 13000 + 455 • 13400 + 50 + 5 = 13400 + 55 • 13450 + 5; and so on.
24 Azim Premji University At Right Angles, March 2021 Divisibility by 10 Any number is divisible by 10 if the last digit of that number is 0. Here we would try to understand the divisibility rule for 10 in the first place as this rule will act as the base to interpret the rules for the other numbers. Observe the way we write the numbers below:
1 11 21 31 41 51 61 71 81 91 101 111 121
2 12 22 32 42 52 62 72 82 92 102 112 122
3 13 23 33 43 53 63 73 83 93 103 113 123
4 14 24 34 44 54 64 74 84 94 104 114 124
5 15 25 35 45 55 65 75 85 95 105 115 125
6 16 26 36 46 56 66 76 86 96 106 116 126
7 17 27 37 47 57 67 77 87 97 107 117 127
8 18 28 38 48 58 68 78 88 98 108 118 128
9 19 29 39 49 59 69 79 89 99 109 119 129
10 20 30 40 50 60 70 80 90 100 110 120 130
Figure 4
All numbers are made of ones and tens where the number of ones must be less than 10 but the number of tens can be as large as we want. For example, 473 is made with 3 ones and 47 tens, while 6850 is made with 685 tens and zero ones. If children have experience in making numbers with bundles (representing tens) and sticks (representing ones), this would be easy for them to understand. Now the tens are all divisible by ten of course but the ones are not. So, the only way a number would be divisible by 10 is if there are no loose ones (i.e., outside the tens) which is when it has 0 as its ones digit (i.e., the last digit).
Divisibility by 5 and 2 A number is divisible by 5 if the last digit of the number is 0 or 5. Similarly, a number is divisible by 2 if the last digit of the number is 0, 2, 4, 6 or 8. One way of looking at these rules can be through the 10 10 grid. × Here, multiples of 2 are highlighted in blue, those of 5 in red and those of both 2 and 5 in purple.
Azim Premji University At Right Angles, March 2021 25 1 11 21 31 41 51 61 71 81 91
2 12 22 32 42 52 62 72 82 92
3 13 23 33 43 53 63 73 83 93
4 14 24 34 44 54 64 74 84 94
5 15 25 35 45 55 65 75 85 95
6 16 26 36 46 56 66 76 86 96
7 17 27 37 47 57 67 77 87 97
8 18 28 38 48 58 68 78 88 98
9 19 29 39 49 59 69 79 89 99
10 20 30 40 50 60 70 80 90 100
Figure 5
A way of justifying these two rules could be through RULE 1 and RULE 2. Any number greater than 10 can be expressed in the form of a sum of a multiple of 10 and the remaining last digit. For example:
In the case of divisibility by 5, for any such number, Part 1 which is a multiple of 10 is always divisible by 5 (RULE 1), and for the whole number to be divisible by 5, the left out last digit should be divisible by 5 (RULE 2). This is possible only if the last digit is either 5 or 0. Hence, the condition for divisibility by5. Can you make a similar argument for divisibility by 2?
Divisibility by 4 A number is divisible by 4 if the number formed by its last two digits is divisible by 4. This is similar to the divisibility rules of 5 and 2. But here, the last two digits of thenumberplayan important role. The reason for this is that while 4 is not a factor of 10, it is a factor of 100. So, inthiscase, Part 1 should be a multiple of 100 (instead of 10), and Part 2 is a 2-digit number whose divisibility it remains to be checked.
26 Azim Premji University At Right Angles, March 2021 For example: Part 1 464 = 400 + 64 = 4 × 100 + 64 → Part 2
Part 1
4596 = 4500 + 96 = 45 × 100 + 96 = 45 × 25 × 4 + 96 → Part 2
Now can you use similar reasoning to understand the rule for divisibility by 8?
Divisibility by 9 and 3 A number is divisible by 9 if the sum of all the digits of the number is divisible by 9. Similarly, a number is divisible by 3 if the sum of all the digits of the number is divisible by 3. Till now, we have been using 10 or 100 as one of the multiples in Part 1 to justify the divisibility rules of 5, 2 and 4. In the case of 9 and 3, Part 1 is written as a multiple of 9, 99, 999, and so on. Let us look through an example: Take a number, say 873, to check its divisibility by 9.
873 = 800 + 70 + 3 = 8 100 + 7 10 + 3 = 8 (99 + 1)+7 (9 + 1)+3 × × × × =(8 99 + 8)+(7 9 + 7)+3 =(8 99 + 7 9)+(8 + 7 + 3) × × × × HerePart1=8 99+7 9 is clearly a multiple of 9. The remaining part or Part 2 i.e. 8 + 7 + 3 =18is × × the sum of the digits of number 873. (Do you see why you get the digits of the given number in Part 2?) So, to check divisibility of 873 by 3 or 9, we need to check only if the digit-sum 18, is divisible by 3 or 9. Since 18 is divisible by 9 (and therefore by 3 also), 873 is divisible by both 9 and 3. Similarly, 4,83,720 = 4 100000 + 8 10000 + 3 1000 + 7 100 + 2 10 + 0 × × × × × = 4 (99999 + 1)+8 (9999 + 1)+3 (999 + 1)+7 (99 + 1)+2 (9 + 1)+0 × × × × × =(4 99999 + 8 9999 + 3 999 + 7 99 + 2 9)+(4 + 8 + 3 + 7 + 2 + 0) × × × × × Again Part 1 is clearly divisible by 9 and Part 2 is the digit-sum = 4 + 8 + 3 + 7 + 2 = 24, which is not divisible by 9 but is divisible by 3. So, 483720 is not divisible by 9, but it is divisible by 3. On the other hand, 5273 = (5 999 + 2 99 + 7 9) + (5 + 2 + 7 + 3) and Part 2 or its digit-sum = 5 + × × × 2 + 7 + 3 = 17 is not divisible by 3 while Part 1 clearly is. So, 5273 is not divisible by 3 or by 9. Now what do you think should be the rule for divisibility by 6? For 12? For 15? For 20?
Azim Premji University At Right Angles, March 2021 27 Some important points to summarise For each divisibility rule, the idea is to break the number into Part 1 and Part 2 such that (i) Part 1 is divisible by the concerned number usually by RULE 1 (ii) Part 2 is very small compared to the original number (iii) We need to check only Part 2 for divisibility and apply RULE 2 for the whole number as depicted below:
Number Part 1 Part 2 2, 5, 10 10m (Multiple of 10) Remaining last digit in units place 4 100m (Multiple of 100) Remaining last 2-digit part 8 1000m (Multiple of 1000) Remaining last 3-digit part 3 and 9 9m (Multiple of 9) Digit-sum
So, we have discussed divisibility by 2, 3, 4, 5, 6 (by extension), 8, 9 and 10. Though we have not proved the rules algebraically for any number, these visualizations will provide enough of a spark to the minds of young learners on why such rules/tests work.
ANKIT PATODI is a Resource Person at Azim Premji Foundation. He has a BE in Computer Science and MA in Development from Azim Premji University, Bangalore. Ankit works with government teachers across various districts of Chhattisgarh on issues related to mathematics and he conducts workshops focusing on conceptual understanding as well as pedagogical strategies used in teaching mathematics. He has been doing mathematics with children for more than 5 years and is interested in exploring and designing teaching resources to address misconceptions in mathematics. Ankit may be contacted at [email protected]
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