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Friday 23 March 2012
DIVISION TRICKS
Division Tricks
divisible by 2:
WHEN THE LAST DIGIT OF ANY NUMBER IS EVEN
(2,4,6,8,0)
Since the last digit of 23578, which is 8, is divisible by 2, then 23578 is
divisible
by 2.
The last digit of 2365 is 5. Since 5 is not divisible
by 2, 2365 is not
divisible
by 2.
divisible by 3:
(WHEN
THE SUM OF ALL NUMBERS IS DIVISIBLE BY 3)
Let us see if 231 is divisible by 3. Summing up the
digits of 231 we get
2 + 3 + 1 = 6. Since 6 is divisible by 3 then 231 is divisible by 3.
Let us check if 23574 is divisible by 3. Summing up the
digits of 23574
gives us
2
+ 3 + 5 + 7 + 4 = 21. Since 21 is divisible by 3, then 23574
is divisible by 3. But suppose that
you could not remember that 21 is divisible by 3.
Then you could sum up the digits of 21 to get 2+1 = 3.
Clearly 3 is divisible by 3. So 21 is divisible by 3 and therefore 23574
is divisible by 3.
Is 27934576592477693217 divisible by 3? Let us sum up
the digits again
2+7+9+3+4+5+7+6+5+9+2+4+7+7+6+9+3+2+1+7 = 105.
If we are not sure if 105 is divisible by
three we can sum up its digits.
So 1 + 0 + 5 = 6. Now we know 6 is divisible by 3, so 105 is divisible
by 3 and therefore 27934576592477693217 is divisible by 3.
The number 16 is not divisible by 3 since 1 + 6 = 7 is not divisible
by
3.
Let us try 23761. It's digits sum to 2 + 3 + 7 + 6 + 1 =
19 and
19 is
not divisible by 3 since 1 + 9 = 10 is not divisible
by 3.
divisible by 4:
(WHEN
LAST 2 DIGITS ARE 00 OR DIVISBLE BY 4)
The number 116 is divisible by 4 since the last two
digits of 116 is 16 and
16 is divisible by 4.
How about the number 1352? The last two
digits of 1352 is 52.
Now 4 will divide 52 (52 _ 4 = 13). So 4 divides 1352.
Will 4 divide 125487100?
Yes since the last two digits are 00.
divisible by 5
(WHEN
NUMBR END WITH 5 OR 0)
It is easy to see that 5 will divide 25, 40, 14520, 785425, and 789875412545,
since each one ends in 0 or 5.
divisible by 6:
(WHEN
THE NUMBER IS divisible by BOTH 2 AND 3)
To check if 132 is divisible by 6 requires two
steps. First is 132 divisible by 2?
Yes since the last digit is divisible by 2.
Next we sum up its digits:1 + 3 + 2 = 6. Since 6 is divisible by 3,
132 is divisible by 3. Therefore 132 is divisible by 6.
Let us try 1764. Clearly 1764 is divisible by 2 and it is
divisible by 3
since 1 + 7 + 6 + 4 = 18 = 9. So 1764 is divisible by 6.
The number 1371 is not divisible by 6 since 1371 is not divisible
by 2.
Note that it is divisible by 3, 1+3+7+1 = 12, but that does
not matter
since it is not divisible by 2.
The number 1678 is not divisible by 6 since it is not
divisible by 3. This
can be seen since 1+6+7+8 = 22. It does not
matter that it is divisible
by 2. It has to be divisible by both.
We can see that 3211 is not divisible
by 6
since
it is not divisible by
either 2 or 3.
divisible by 7:
Let us start with some easy examples.
First 14 is divisible by 7 since
1 - 2(4) = 1 - 8 = -7 and -7 is divisible by 7.
Next 21 is divisible by
7 since 2 - 2(1) = 2 - 2 = 0 and 0 is divisible by 7.
Is 924 divisible by 7?
We apply the rule to get 92-2(4) = 92-8 = 84.
If we are not sure whether 7 divides 84 we can apply the
rule again.
2
Doing this gives us 8-2(4) = 8-8 = 0 and 0 is divisible by 7. So 924
is divisible by 7.
Our next example is to determine if 2632 is divisible by 7.
Apply
the rule we get 263 - 2(2) = 263 - 4 = 259.
Applying the rule again
25 - 2(9) = 25 - 18 = 7. So 7 divides 2632.
Is 7532 divisible by 7? Let us apply the rule: 753-2(2) = 753-4 = 749.
Applying the rule again we get 74 - 2(9) = 74 - 18 = 56. If we do not
recognize that 56 = 7 _ 8 we can apply the rule another time to get
5 - 2(6) = 5 - 12 = -7. So 7 divides 7532.
Does 7 divide 72? Well 7 - 2(2) = 7 - 4 = 3, so 7 does not divide 72.
divisible by 8:
(WHEN
LAST 3 DIGITS ARE 000 OR DIVISBLE BY 8)
This is a little tricker to apply since you
have to look at the last three digits.
For example, does 8 divide 3120? Since 8 divides 120, 8 divides
3120. But you have to know that 8 divides 120 to apply this
trick. So
normally I just divide the number by 2 and if I can do
this three times
the number is divisible by 8. Of course I
would already know that since
I have the quotient. There are situations
where this rule is actually
useful for me.
First if the number ends in three 0's I know it is
divisible by 8. For
example, the number 1548000 is divisible by 8. The other time I
_nd
this trick useful is if the third digit is a 0.
Then I can just check to see
if the last two digits are divisible by 8. As an example
let us look at
547024. To check this number for divisibility by 8 we look at 024 or as
we usually write it 24. Since 24 is divisible by 8, so is 547024.
divisible by 9:
Is 234 divisible by 9? Yes since 2+3+4 = 9 and 9 is divisible by 9.
Also
31275 is divisible by 9 since 3 + 1 + 2 + 7 + 5 =
18 and
18 is divisible
by 9 (1 + 8 = 9).
Let us try 573. Since 5 + 7 + 3 = 15 and 15 is not divisible by 9, 573
is not divisible by 9.
divisible by 10:
The following are all divisible by 10 since they end in
a 0: 20, 25640,
87951880,and 603020105040708090201020.
divisible by 11:
(SUM
OF ALL NUMBER ON ODD POSITION - SUM OF ALL NUMBER ON EVEN POSITION)
Does 11 divide 143?
To check we do the following calculation: (1+3)-
4 = 4 - 4 = 0. Since 11 divides 0 it divides 143.
To check to see if 11 divides 2607 we make the
calculation: (2 + 0) -
(6 + 7) = 2 - 13 = -11. Since 11 divides -11 it divides 2607.
Now try 38236. So (3 + 2 + 6) - (8 + 3) = 11 - 11 = 0. So 11 divides
38236.
Finally does 11 divide 3042149? Well since 11 divides (3+4+1+9)-
(0 + 2 + 4) = 17 - 6 = 11 it divides 3042149.
DIVISIBLE BY12:
(it
has to be divisible by both 3 and 4)
Is 132 divisible by 12? If it is it has to be divisible by both 3 and 4. Since
1 + 3 + 2 = 6 it is divisible by 3 and since 32 is divisible by 4, 132 is
divisible by 12.
What about 264? Again we test for divisibility by 3 and 4. Since 2 +
6 + 4 = 12 is divisible by 3 and 64 is divisible by 4, 264 is divisible by
12.
Can we divide 1800 by 12? First we add the digits of 1800 to get
1+8+0+0 = 9. So 1800 is divisible by 3. The last two
digits of 1800
are 00, so 1800 is divisible by 4. Therefore 1800 is divisible by 12.
DIVISIBLE BY 13:
Is 65 divisible by 13? By the rule we compute
6 - 9(5) = 6 - 45 = -39:
Since 39 is divisible by 13, 65 is divisible by 13.
Lets try 377.
37 - 9(7) = 37 - 63 = -26
So 377 is divisible by 13, since 26 is divisible by 13.
Is 2041 divisible by 13?
204 - 9(1) = 195
19 - 9(5) = 19 - 45 = -26
So 2041 is divisible by 13.
BY: AJAY JINDAL
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