The digits add up to 57, and 5 plus seven is 12, so the original number is divisible by three.. is to look at its [ones digit]: if it is even, and the sum of the [digits] is a multiple
Trang 1Multiply Up to 20X20 In Your Head
In just FIVE minutes you should learn to quickly multiply up to 20x20 in your head With this trick, you will be able to multiply any two numbers from 11
to 19 in your head quickly, without the use of a calculator
I will assume that you know your multiplication table reasonably well up to 10x10
Try this:
• Take 15 x 13 for an example
• Always place the larger number of the two on top in your mind
• Then draw the shape of Africa mentally so it covers the 15 and the 3 from the 13 below Those covered numbers are all you need
• First add 15 + 3 = 18
• Add a zero behind it (multiply by 10) to get 180
• Multiply the covered lower 3 x the single digit above it the "5" (3x5= 15)
• Add 180 + 15 = 195
That is It! Wasn't that easy? Practice it on paper first!
The 11 Rule
You likely all know the 10 rule (to multiply by 10, just add a 0 behind the number) but do you know the 11 rule? It is as easy! You should be able to do this one in you head for any two digit number Practice it on paper first!
To multiply any two digit number by 11:
• For this example we will use 54
• Separate the two digits in you mind (5 4)
• Notice the hole between them!
• Add the 5 and the 4 together (5+4=9)
• Put the resulting 9 in the hole 594 That's it! 11 x 54=594
The only thing tricky to remember is that if the result of the addition is greater than 9, you only put the "ones" digit in the hole and carry the "tens" digit from the addition For example 11 x 57 5 7 5+7=12 put the 2 in the hole and add the 1 from the 12 to the 5 in to get 6 for a result of 627 11 x 57 = 627
Practice it on paper first!
Finger Math: 9X Rule
To multiply by 9,try this:
(1) Spread your two hands out and place them on a desk or table in front of you (2) To multiply by 3, fold down the 3rd finger from the left To multiply by 4, it would be the 4th finger and so on
(3) the answer is 27 READ it from the two fingers on the left of the folded down finger and the 7 fingers on the right of it
This works for anything up to 9x10!
Trang 2Square a 2 Digit Number Ending in 5
For this example we will use 25
• Take the "tens" part of the number (the 2 and add 1)=3
• Multiply the original "tens" part of the number by the new number (2x3)
• Take the result (2x3=6) and put 25 behind it Result the answer 625
Try a few more 75 squared = 7x8=56 put 25 behind it is 5625
55 squared = 5x6=30 put 25 behind it is 3025 Another easy one! Practice it
on paper first!
Square 2 Digit Number: UP-DOWN Method
Square a 2 Digit Number, for this example 37:
• Look for the nearest 10 boundary
• In this case up 3 from 37 to 40
• Since you went UP 3 to 40 go DOWN 3 from 37 to 34
• Now mentally multiply 34x40
• The way I do it is 34x10=340;
• Double it mentally to 680
• Double it again mentally to 1360
• This 1360 is the FIRST interim answer
• 37 is "3" away from the 10 boundary 40
• Square this "3" distance from 10 boundary
• 3x3=9 which is the SECOND interim answer
• Add the two interim answers to get the final answer
• Answer: 1360 + 9 = 1369
Multiply By 4
To quickly multiply by four, double the number and then double it again Often this can be done in your head
Multiply By 5
To quickly multiply by 5, divide the number in two and then multiply it by 10 Often this can be done quickly in your head
Trang 3The 11 Rule Expanded
You can directly write down the answer to any number multiplied by 11
• Take for example the number 51236 X 11
• First, write down the number with a zero in front of it
051236
The zero is necessary so that the rules are simpler
• Draw a line under the number
• Bear with me on this one It is simple if you work through it slowly To do this, all you have to do this is "Add the neighbor" Look at the 6 in the "units" position of the number Since there is no number to the right of it, you can't add to its "neighbor" so just write down 6 below the 6 in the units col
• For the "tens" place, add the 3 to the its "neighbor" (the 6) Write the answer: 9 below the 3
• For the "hundreds" place, add the 2 to the its "neighbor" (the 3) Write the answer: 5 below the 2
• For the "thousands" place, add the 1 to the its "neighbor" (the 2) Write the answer: 3 below the 1
• For the "ten-thousands" place, add the 5 to the its "neighbor" (the 1) Write the answer: 6 below the 5
• For the "hundred-thousands" place, add the 0 to the its "neighbor" (the 5) Write the answer: 5 below the 0
That's it 11 X 051236 = 563596
Practice it on paper first!
Trang 4Divisibility Rules
Dividing by 3
Add up the digits: if the sum is divisible by three, then the number is as
well Examples:
1 111111: the digits add to 6 so the whole number is divisible by three
2 87687687 The digits add up to 57, and 5 plus seven is 12, so the original number is divisible by three
Why does the 'divisibility by 3' rule work?
From: Dr Math To: Kevin Gallagher Subject: Re: Divisibility of a number by 3
As Kevin Gallagher wrote to Dr Math
On 5/11/96 at 21:35:40 (Eastern Time),
>I'm looking for a SIMPLE way to explain to several very bright 2nd
>graders why the divisibility by 3 rule works, i.e add up all the
>digits; if the sum is evenly divisible by 3, then the number is as well
>Thanks!
>Kevin Gallagher The only way that I can think of to explain this would be as follows: Look
at a 2 digit number: 10a+b=9a+(a+b) We know that 9a is divisible by 3,
so 10a+b will be divisible by 3 if and only if a+b is Similarly, 100a+10b+c=99a+9b+(a+b+c), and 99a+9b is divisible by 3, so the total will be iff a+b+c is
This explanation also works to prove the divisibility by 9 test It clearly originates from modular arithmetic ideas, and I'm not sure if it's simple enough, but it's the only explanation I can think of
Doctor Darren, The Math Forum Check out our web site - http://mathforum.org/dr.math/
Dividing by 4
Look at the last two digits If the number formed by its last two digits is divisible by 4, the original number is as well
Examples:
3 100 is divisible by 4
4 1732782989264864826421834612 is divisible by four also, because 12 is divisible by four
Dividing by 5
Trang 5If the last digit is a five or a zero, then the number is divisible by 5
Dividing by 6
Check 3 and 2 If the number is divisible by both 3 and 2, it is divisible by
6 as well
Robert Rusher writes in:
Another easy way to tell if a [multi-digit] number is
divisible by six
is to look at its [ones digit]: if it is even, and the sum
of the [digits] is
a multiple of 3, then the number is divisible by 6.
Dividing by 7
To find out if a number is divisible by seven, take the last digit, double it, and subtract it from the rest of the number
Example: If you had 203, you would double the last digit to get six, and
subtract that from 20 to get 14 If you get an answer divisible by 7
(including zero), then the original number is divisible by seven If you don't know the new number's divisibility, you can apply the rule again Matthew Correnti describes this method:
If you do not know if a two-digit number, call it ab, is divisible
by 7, calculate 2a + 3b This will yield a smaller number, and if
you do the process enough times you will eventually if the
number ab is divisible by 7 end up with 7.
You can use a similar method if you have a three-digit number abc:
take the digit a and multiply it by 2, then add it to the number bc,
giving you 2a + bc; repeat and reduce until you recognize the
result's divisibility by seven With a four-digit number abcd, take
the digit a and multiply by 6, then add 6a to bcd giving This
usually gives you a three-digit number; call it xyz Take that x
and, as described previously, multiply x by two and add to
yz
(i.e., 2x + yz) Again, repeat and reduce until you
recognize the
result's divisibility by seven.
Ahmed Al Harthy writes in:
To know if a number is a multiple of seven or not, we can use also
3 coefficients (1 , 2 , 3) We multiply the first number starting
from the ones place by 1, then the second from the right by
3,
the third by 2, the fourth by -1, the fifth by -3, the sixth
by -2,
and the seventh by 1, and so forth.
Example: 348967129356876
Trang 66 + 21 + 16 - 6 - 15 - 6 + 9 + 6 + 2 - 7 - 18 - 18 + 8 + 12 + 6 = 16
means the number is not multiple of seven.
If the number was 348967129356874, then the number is a multiple of seven
because instead of 16, we would find 14 as a result, which
is a multiple of 7
So the pattern is as follows: for a number onmlkjihgfedcba, calculate
a + 3b + 2c - d - 3e - 2f + g + 3h + 2i - j - 3k - 2l + m + 3n + 2o.
Example: 348967129356874.
Below each digit let me write its respective figure.
3 4 8 9 6 7 1 2 9 3 5 6 8 7 6
2 3 1 -2 -3 -1 2 3 1 -2 -3 -1 2 3 1
(3×2) + (4×3) + (8×1) + (9×-2) + (6×-3) + (7×-1) +
(1×2) + (2×3) + (9×1) + (3×-2) + (5×-3) + (6×-1) +
(8×2) + (7×3) + (6×1) = 16 not a multiple of seven.
Another visitor observes:
Here is one formula for seven
3X + L
L = last digit
X = everything in front of last digit.
All numbers that are divisible by seven have this in common There are no exceptions.
For example, 42: 3(4) + 2 = 14.
Seven divides into 14, so it divides into 42.
Next example, 105: 3(10) + 5 = 35.
Seven divides into 35, so it divides into 105.
Here is another formula for seven:
4X - L
When using this formula, if you get zero, seven or a
multiple of seven,
the number will be divisible by seven.
For example, 56: 4(5) - 6 = 14.
Seven divides into 14, so it divides into 56.
Next example, 168: 16(4) - 8 = 56.
Seven divides into 56, so it divides into 168.
Similarly:
The formula for 2 is 2X + L
The formula for 3 is 4X + L
The formula for 4 is 6X + L
Trang 7The formula for 5 is 5X + L
The formula for 6 is 2X + L and 4X + L in other words, the formulas for 2 and 3
must work before the number is divisible by 6.
The formula for 9 is X + L
The formula for 11 is X - L
The formula for 12 is 2X - L
The formula for 13 is 3X - L
The formula for 14 is 4X - L and 2X + L in other words, the formulas for 7 and 2
must work before the number is divisible by 14.
The formula for 17 is 7X - L
The formula for 21 is X - 2L
The formula for 23 is 3X - 2L
The formula for 31 is X - 3L
Sara Heikali explains this way to test a number with three or more digits for divisibility by seven:
1 Write down just the digits in the tens and ones places.
2 Take the other numbers to the left of those last two digits,
and multiply them by two.
3 Add the answer from step two to the number from step one.
4 If the sum from step three is divisible by seven, then the
original number is divisible by seven, as well If the sum
is
not divisible by seven, then the original number is not divisible by seven.
For example, if the number we are testing is 112, then
1 Write down just the digits in the tens and ones places: 12.
2 Take the other numbers to the left of those last two digits,
and multiply them by two: 1 × 2 = 2.
3 Add the answer from step two to the number from step one:
12 + 2 = 14.
4 Fourteen is divisible be seven Therefore, our original number, one hundred twelve, is also divisible by seven.
See also Explaining the Divisibility Rule for 7 in the Dr Math archives
Dividing by 8
Check the last three digits Since 1000 is divisible by 8, if the last three digits of a number are divisible by 8, then so is the whole number
Example: 33333888 is divisible by 8; 33333886 isn't
How can you tell whether the last three digits are divisible by 8? Phillip McReynolds answers:
If the first digit is even, the number is divisible by 8 if the last two digits are If the first digit is odd, subtract 4 from the last two digits; the number will be divisible by 8 if the resulting last two digits are So, to continue the
last example, 33333888 is divisible by 8 because the digit in the hundreds
place is an even number, and the last two digits are 88, which is divisible
by 8 33333886 is not divisible by 8 because the digit in the hundreds
Trang 8place is an even number, but the last two digits are 86, which is not
divisible by 8
Sara Heikali explains this test of divisibility by eight for numbers with three or more digits:
1 Write down the units digit of the original number.
2 Take the other numbers to the left of the last digit, and multiply them by two.
3 Add the answer from step two to the number from step one.
4 If the sum from step three is divisible by eight, then the
original number is divisible by eight, as well If the sum
is
not divisible by eight, then the original number is not divisible by eight.
For example, if the number we are testing is 104, then
1 Write down just the digits in ones place: 4.
2 Take the other numbers to the left of that last digit, and multiply them by two: 10 × 2 = 20.
3 Add the answer from step two to the number from step one:
4 + 20 = 24.
4 Twenty-four is divisible be eight Therefore, our
original
number, one hundred and four, is also divisible by eight.
Dividing by 9
Add the digits If that sum is divisible by nine, then the original number is
as well
Dividing by 10
If the number ends in 0, it is divisible by 10
Dividing by 11
Let's look at 352, which is divisible by 11; the answer is 32 3+2 is 5;
another way to say this is that 35 -2 is 33
Now look at 3531, which is also divisible by 11 It is not a coincidence
that 353-1 is 352 and 11 × 321 is 3531
Here is a generalization of this system Let's look at the number 94186565.
First we want to find whether it is divisible by 11, but on the way we are going to save the numbers that we use: in every step we will subtract the last digit from the other digits, then save the subtracted amount in order Start with
9418656 - 5 = 9418651 SAVE 5
Then 941865 - 1 = 941864 SAVE 1
Then 94186 - 4 = 94182 SAVE 4
Then 9418 - 2 = 9416 SAVE 2
Then 941 - 6 = 935 SAVE 6
Then 93 - 5 = 88 SAVE 5
Trang 9Then 8 - 8 = 0 SAVE 8
Now write the numbers we saved in reverse order, and we have 8562415, which multiplied by 11 is 94186565.
Here's an even easier method, contributed by Chis Foren:
Take any number, such as 365167484.
Add the first, third, fifth, seventh, , digits 3 + 5 + 6 + 4 + 4 = 22 Add the second, fourth, sixth, eighth, , digits 6 + 1 + 7 + 8 = 22
If the difference, including 0, is divisible by 11, then so is the number
22 - 22 = 0 so 365167484 is evenly divisible by 11.
See also Divisibility by 11 in the Dr Math archives
Dividing by 12
Check for divisibility by 3 and 4
Dividing by 13
Here's a straightforward method supplied by Scott Fellows:
Delete the last digit from the given number Then subtract nine times the deleted digit from the remaining number If what is left is divisible by 13, then so is the original number
Rafael Ando contributes:
Instead of deleting the last digit and subtracting it ninefold from the remaining number (which works), you could also add the deleted digit fourfold Both methods work because 91 and 39 are each multiples of 13
For any prime p (except 2 and 5), a rule of divisibility could be "created" using this method:
1. Find m, such that m is the (preferably) smallest multiple of p that ends in either 1
or 9
2. Delete the last digit and add (if multiple ends in 9) or subtract (if it ends in 1) the deleted digit times the integer nearest to m/10 For example, if m =
91, the integer closest to 91/10 = 9.1 is 9; and for 3.9, it's 4
3. Verify if the result is a multiple of p Use this process until it's obvious
Example 1: Let's see if 14281581 is a multiple of 17
In this case, m = 51 (which is 17×3), so we'll be deleting the last number and subtracting it fivefold
Trang 101428158 - 5×1 = 1428153
142815 - 5×3 = 142800
14280 - 5×0 = 14280
1428 - 5×0 = 1428
142 - 5×8 = 102
10 - 5×2 = 0, which is a multiple of 17, so 14281581 is multiple of 17
Example 2: Let's see if 7183186 is a multiple of 46
First, note that 46 is not a prime number, and its factorization is 2×23 So,
7183186 needs to be divisible by both 2 and 23 Since it's an even number, it's obviously divisible by 2
So let's verify that it is a multiple of 23:
m = 3×23 = 69, which means we'll be adding the deleted digit sevenfold
718318 + 7×6 = 718360
71836 + 7×0 = 71836
7183 + 7×6 = 7225
722 + 7×5 = 757
75 + 7×7 = 124
12 + 7×4 = 40
4 + 7×0 = 4 (not divisible by 23), so 7183186 is not divisible by 46
Note that you could've stopped calculating whenever you find the result to be obvious (i.e., you don't need to do it until the end) For example, in example 1 if you recognize 102 as divisible by 17, you don't need to continue (likewise, if you recognized 40 as not divisible by 23)
The idea behind this method it that you're either subtracting m×(last digit) and then dividing by 10, or adding m×(last digit) and then dividing by 10
Jeremy Lane adds:
It may be noted that while applying these rules, it is possible to loop among numbers as results
Example: Is 1313 divisible by 13?
Using the procedure given we take 13×3 and obtain 39 This multiple ends in 9 so
we add four-fold the last digit
131 + 4×3 = 143
14 + 4×3 = 26
2 + 4×6 = 26
Example: Is 1326 divisible by 13?
Using the procedure given we take 13×7 = 91 This is not the smallest multiple, but it does show looping The smaller multiple does loop at 39 as well There are some examples where we would still need to recognize certain multiples So we subtract nine-fold the last digit