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DIVISIBILITY WORK SHEET
i)
Divisibility by 2, this is straight forward, if the number is even then it will be
divisible by 2
Question: Show that 46 is divisible by 2 but 159 is not
Generalization about dividing by 2:
ii)
Divisibility by 3: Which of the following are divisible by 3,
42, 56, 27, 52, 255,
The rule is:_______________________________________________
Initially we will have to examine what happens when powers of 10 are divided by 3
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Use this factoring of powers of ten to demonstrate why 252 is divisible by 3
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Why and how is 1461 divisible by 3
Generalization about dividing by 3:
iii)
divisibility by 4: Which of the following are divisible by 4
34, 132, 256, 1348,
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Before we go for a general rule have a think about powers of 10 divided by 4
So what general statement can we make about division by 4
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Consider the divisibility of 1456 by 4
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Write out an explanation of why 5436 is divisible by 4
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Generalization about dividing by 4:
iv)
Divisibility by 5: This one is very straight forward so we can go straight to the
generalization
Generalization about dividing by 5:
Give a full explanation as to why 8450 is divisible by 5
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v)
Divisibility by 6: By considering carefully the multiples of 6 check to see if you
can find a rule for divisibility by 6.
Use your rule to decide which of the following are divisible by 6
532552,44 135,767,890
373,855,056
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Now give a full explanation of how your rule works showing why 1446 is divisible
by 6
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Generalization about dividing by 6:
vi)
Divisibility by 7!. This is an awkward one so I will guide you through an
algorithm for divisibility by 7 . Consider 25,032÷7
From 25032 form the number 2503 – 2x2=2499
Doing the same thing again to get 249-2x9=231
Again gives
23-2x1=21 but 21 is divisible by 7 so then is 231
and
2499 and so is 25,032 .
Use the same algorithm to establish that 17,395 is devisable by 7
I will present a proof at the end of this investigation but it will be a bit beyond
what pupils can be expected to grasp.
Check which of the following are divisible by 7
245, 1946, 4875, 394,975.
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Generalization about dividing by 7:
vii)
Divisibility by 8: Now we know that 1000 is divisible by 8 so then are all
multiples of 1000. that means we only really have to concern ourselves with the
first three digits of any number.
Check out which of the following numbers is divisible by 8
1,144 1,736 3,776 9,872 38,172
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Is there a Rule:________________________________________________________
Use your rule to explain why 27,832 is divisible by 8
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Generalization about dividing by 8
viii) Divisibility by 9: A quick reflection on the multiples of 9 should give us a rule to
test for dividing. Using the following 10 = 9 + 1, and that 100 = 99 + 1, 1000= 999 + 1
etc. explain fully why 24,669 is divisible by 9
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Generalization about dividing by 9:
viii)
Divisibility by numbers greater than 9: 10 is straight forward and easy to
validate. We will finish off with an examination of how to test for divisibility by
11.
The following are all divisible by 11. Examine the values and see if there is a
generalization we can use.
13,541:____________________________________
27,357:____________________________________
358,149:___________________________________
471,625___________________________________
1,261,953:_________________________________
Generalization about dividing by 11:
I have not examined any further than this but there will be algorithms for division by other
values!!
‘Much of mathematics consists of attempts to replace difficult problems by easier ones having the same answer’
Most high school pupils
Proof of Divisibility by 7:
I have used a proof from D. Holton which goes
Let N represent be any number written as
N= an an1an2 ..........a2 a1a0
And let M= an an1an2 .........a2 a1  2 xa0
Now N is divisible by 7 if and only if M is divisible by 7
To complete this theorem we will put L= an an1an2 ..........a2 a1
 N=10l+ a 0 and M= L- 2a 0
If N is divisible by 7 then so is 2n = 20L+2 a 0
Now obviously 21L + 7a 0 is divisible by 7
But then ( 21L + 7a 0 )—(20L+ 2a 0 ) is also divisible by 7
but ( 21L + 7a 0 )—(20L+ 2a 0 )= L+ 5a0
And likewise L+ 5a0 - 7a 0 must also be divisible by 7
But L+ 5a0 - 7a 0 +=M
Now If M is divisible by 7 then so is N
If L- 2a 0 is divisible by 7 then so is 10(L- 2a 0 ) = 10L-20 a 0
But 21a0 must also be divisible by 7 then so must
10L-20 a 0 +21 a 0 =10L+ a 0 =N
So N is divisible by 7
PROOF THAT 2 X 2 = 5!!
LetA  4
B5
C 1
ThenC  ( B  A)
soC ( B  A)  ( B  A) 2
CB  CA  B 2  2 AB  A 2
subtractA 2
CB  CA  A 2  B 2  2 AB
addAB
AB  CB  CA  A 2  B 2  AB
subtractCB
AB  CA  A 2  B 2  AB  CB
A( B  C  A)  B ( B  A  C )
A B
so 4  5
so 2 X 2  5!!
Algebraic symbols are used when you don’t know what you are talking about’
Anonymous