Finding Perfect Squares

Now if we find a number such as √225 we know that it is 15 if we memorized our tables.

What if we run into a very large number like 21316?  If we were told find the root without a calculator, it would be very hard you would think.  We need to come up with a quick formula for solving this.

Lets break this down into a formula.  We will create an equation for the root by placing A in the 10's place and B in the units place.  We will square this number to come up with an equation for solving the root.

10A+B= root
(10A +B)²=21316

100A² + 20AB + B²=21316

Finding A:

• Before finding A, I will subtract 16 from 21316: 21316-16=21300
• Divide 21300 by 100: 21300÷100=213
• We will find the nearest perfect square of a number less than 213 and let this number =A.  That would be 14 because 14²=196.  We cant use 15 because 15²=225 which is larger than 213.  So A=14.

Finding B:

• We will take 100A²=100X14²= 19600 and subtract it from or original number:  21316-19600=1716
• Our equation becomes 20AB+B²=1716 or if we substitute A back into our equation we have 20X14XB+B²=280B+B².  Because the number ends in a 6 we know that the units digit or B is either 4 or 6.  4 does not work because 280(4)+4²=1136≠1716.  If we substitute 6 into our equation, we have 280(6)+36=1716.

Voila our number is 156.  15 is in the tens place and 6 is in the units place.

It is not really that hard.  We will do an easier number 841.

If we subtract 41 and divide by 100, we have 8 and we have 2²=4 which is the largest perfect square less than 8. A=2 so 100A²=400.  20AB+B²=841-400=441.  So 20AB=20(2)B+B² =40B+B²=441.
B can be either 1 or 9.  B is 9.  The square root of 841 is 39.

 Perfect Square Rules

Lets start with the simple rules previously discussed.

Any perfect square that ends in a 1 has a root of a number that ends in 9 or 1.

21²=441

19²=361

Any perfect square that ends in a 4  has a root of a number that ends in 2 or 8.

22²=484

18²=324

Any perfect square that ends in a 9  has a root of a number that ends in 3 or 7.

17²=289

33²=1089

Any perfect square that ends in a 0  has a root of a number that ends a 0.

20²=400

Any perfect square that ends in a 5  has a root of a number that ends a 5.

25²=625

Any number ending in 2,3,7, or 8 is not a perfect square.

√23 is not a perfect square
√38 is not a perfect square

Every perfect square except 1 is either a multiple of 3 one more than a multiple of 3:
25=8x3+1

169=56x3+1

144=48x3

225=75x3

Every perfect square except 1 is either divisible by 4 or is 1 more that a multiple of 4:

25=4x6+1

256=4x64

289=4x72+1

Every perfect square whose units digit is 5 always ends in 25. 

15²=225

215²=46225

I never thought rules would be so cool.

Decimals to % and % to Decimals

I teach my students to count by 2 to turn a decimal to a percentage.

Converting decimals to percentages:

To convert a decimal into a percentage, we move the decimal point 2 digits to the right of the number and add a % sign.

Examples:

.12 =12% 

.132=13.2%

.25=25%

0.014=1.4%

4.29=429%

Converting percentages to decimal:

To convert a percentage into a decimal, we move the decimal point 2 digits to the left of the number and remove the % sign.

Examples:

30%=.30
41.7%=.417

125%=1.25

.34%=.0034

Now try for yourself:

Convert 21%, 2.3%, 60.1% 234% into decimals
Convert .34, .75, 1.39, .045 into percentages

Answers: .21, .023, .601, 2.34, 34%, 75%, 139%, and 4.5%

Fast Math Squaring Numbers Ending in 1

There are a couple of real nifty ways of doing fast math with numbers that end in 1.  We are going to present a addition method.  There are other methods that we will present in subsequent posts but this method presented is the easiest way to calculate squares ending in 1.  

Addition method:
Lets start with a number 71.

• We will subtract 1 from this number and call it our base number: 71-1=70
• We will square our base number: 70²=4900
• We will multiply our base number by 2: 70x2=140
• We will add the square of our base number to our base number multiplied by 20 and add 1.  4900+140+1=5041

That is pretty easy isn't it?

Lets find 101²
101-1=100

100²=10,000
2X100=200
10,000+200+1=10,201 

That is incredibly easy and fast!  Lets try one more number:

191²

191-1=190
190²=36100

2X190=380
36100+380+1=36481

Outstanding!  Now you know how to square numbers ending in one and you can do it very fast.  We will show how to do other numbers in subsequent posts so stay tuned.

Fast Math Squaring Numbers That End in 5

Squaring numbers that end in 5 are very easy and can be done very fast.

The fast math tricks have actually many different ways to solve but I will present 3.

1) Addition method:

Lets start with a number 55.
Subtract 5 from the original number: 55-5=50.

We will call this number our base number.

Square the base number: 50²=2500

Multiply the base number by 10: 10x50=500

Add the square of the base number to the base number x 10 and add 25: 2500+500+25=3025

If we wanted to find out what 115² equals, we would do the following:
115-5=110
110²=12100

110x10=1100
115²=12,100+1100+25=13,225

2) Subtraction method:

Lets start with a number 55 again.
Add 5 from the original number: 55+5=60.

We will call this number our base number.

Square the base number: 60²=3600

Multiply the base number by 10: 10x60=600

Subtract the base number x 10 from the base number squared and add 25: 3600-600+25=3025

Pretty cool so far right?

Lets use this methodology to find 95²

95+5=100
100²=10,000

100x10=1000
95²=10,000-1000+25=9025

3) Multiplication method

Lets start with a number 55 again.

Add 5 from the original number: 55+5=60.

We will call this number our base number 1.

Subtract 5 from the original number: 55-5=50

We will call this base 2.

Multiply base 1 to base 2 and add 25: 50x60+25=3025

Wow!  That is really amazing!

Lets try 205²
205-5=200

205+5=210
200X210=42000
42000+25=42025

In later post, I will post how these apply to squares of other numbers.

If you practice and memorize 2 or 3 digit number multiples, you can multiply really fast. I will post some practice on doing these with each unit number digit and you will get really fast and impressive and you will do some difficult problems quickly on standardized tests like the SAT.

Simple Rules for Perfect Squares

Lets look at the first 10 perfect squares and we will observe some properties.

1² =1

2² =4

3² =9

4² =16

5² =25

6² =36

7² =49

8² =64

9² =81

10² =100

0 through 9 represent the units digits of all numbers.  

Our numbers show: 

All numbers that end in 1 or 9 will produce a units digit of 1.

All numbers that end in a 2 or a 8 will produce a units digit of 4.

All numbers that ends in a 3 or a 7 will produce a units digit of 9.

All numbers that end in 5 will produce a units digit of 5.

All numbers that end in 0 will produce a units digit of 0.

We know that any number that ends in 2, 3, 7, or 8 does not have a perfect square.

32, 87,63, 1028 are not perfect squares.

In all perfect squares, they have twice the number of zeros as the number being squared.  All perfect squares have an even number of zeros.

20, 3,000, 40,000 are not perfect squares.

60²=3600 (2 zeros)
1100²=1,210,000 (4 zeros)

49000²=2,401,000,000 (6 zeros)

Exercises:

Which of the following could be perfect squares based on the rules we presented.

121, 310, 278, 6400, 497, 1,000, 2,500, 529

Perfect square is the square of an integer. 3² =9 so 9 is a perfect square.  20 is the result of (√20)² and √20 does not produce a integer therefore 20 is not a perfect square.

Fast Squares Ending in 9

Easily square numbers in your head like 109² and 69².  I will show you how and you will impress your friends. 

How would you like to be able to do squares really fast in your head?  This is the quick trick and it is really easy.  Lets say someone asks you what 69 times 69 is and you immediately say 4761.  You might be asking, "How did you do that?"

Okay here we go: 70²−2(70)+1

Okay, I heard some groans after that.  I will explain this methodology and it is really easy.  I promise.

Steps for squaring a number that ends in 9: 
Add 1 to the number so that the number ends in a 0.
Lets go back to 69.  1+69=70

Square the number we added 1 to.
70²=4900

Multiply by 2 the number we added 1 to.
2(70)=140

Subtract the multiplied number from the squared number and add 1.
4900-140+1=4761

This works with any number that ends in 9.  Here is why it works.

If we have a number 70², we know that 69x70 is 70 less than 70².  69x69 is 69 less than 69x70 so if we wanted to know what 69² is we could simply take 70² and subtract 70 and subtract 69.  That is kind of hard to do in our head so lets do the following:

-(69+70)=-(70 +70 -1) This can be re-written as -2(70) +1
So Voila we have 69²=70²−2(70)+1

Lets try 109²

We add 1 and have 110. 
110²=12100
2(110)=220
12100-220+1=11881

Phew!  Pretty nice isn't it?  But what about numbers that end in an 8 or 1 or what about all the other numbers?  We will learn how to square all the numbers in following posts.

Divisibility Rules

There are some special rules that make multiplying, factoring and dividing a lot easier.

The first rule is that any number that is divisible by 2 has to be even and if the number is even the number is divisible by 2.

8 is even so it is divisible by 2
8÷2=4
22 is even so it is divisible by 2
22÷2=11
23 is odd therefore it is not divisible by 2.
23÷2=11.5 

Any number that is divisible by 10 will end in a zero.  If a number ends in a zero, it is divisible by 10.

130 ends in a 0 so it is divisible by 10
130÷10=13
122 does not end in a zero so it is not divisible by 10
122÷10=1.22

Any number that is divisible by 5 will end in 5 or a 0.  If a number ends in 5 or 0, it is divisible by 5.

125 ends in a 5 so it is divisible by 5
125 ends in 5 so it is divisible by 5
125÷5=25
160 ends in 0 so it is divisible by 5
160÷5=32
201 does not end in 5 or 0 so it is not divisible by 5.
201÷5=40.2

If a number that is divisible by 3, then the sum of its digits is divisible by 3.  If the digits of a number add up to something other than 3, then it is not divisible by 3.

114 is a number whose sum of digits equals 6 which is divisible by 3 so 114 is divisible by 3.
114÷3=38
514 is a number whose sum of digits equals 10 which is not divisible by 3 so 114 is not divisible by 3.
514÷3=171.333333....

If a number that is divisible by 4, then the last 2 digits are divisible by 4.  If the last 2 digits of a number are not divisible by 4, then the number is not divisible by 4.

724 is a number such that the last 2 digits that are divisible by 4 so 724 is divisible by 4.
724÷4=181
718 is a number such that the last 2 digits that are not divisible by 4 so 724 is not divisible by 4.
718÷4=179.5

Exercises:

Find out if the following numbers are divisible by 2, 3, 5, or 10 or not divisible by any of the numbers.

25, 2864, 1290, 441, 1236, 445, 221, 708, and 433.

Fast Multiplication

When students learn the multiplication tables in 3rd grade (I think that is when they do it), they memorize numbers up to a 10x10 matrix and sometimes a little higher. There was a time that it was a need for people to be able to do basic arithmetic in their heads.  Unfortunately people now rely on calculators which means that people do not gain the ability to have a feel for numbers. When a number is higher than this 10x10 matrix, many people cannot multiply it in their heads.   

To do the following calculations, I assume that a person has a knowledge of all the multiples in a 10 by 10 matrix.

Lets start with all the squares. 11x11=
Here is how to do it in your head:

1x11=11 and 10 x 11 =110
so 11+110=121

Not too bad but some may not see this at first so lets discuss it. We have a 2 digit number 11 and it is composed of a 10's digit and units digit. If we break 11 into 10 and 1 it is easy to multiply and fairly easy to add in our heads. The part that is hard when multiplying numbers in our heads is remembering all the numbers and placing the numbers in their corresponding slots (ie thousands, hundreds, tens, and units).  The more digits we have the harder it becomes to keep track of these numbers in our head.  We need something to help us. 

I will expaln this as we multiply 12 x 12. How hard is it to multiply 12 by 10? It is not hard at all. How hard is it to remember? Not hard at all. 12 x 12 is equal to 24 + 120=144. What you are doing is multiplying 2 by 12 and getting 24 and knowing that you will add this to 120.

So lets do some more:

13x13=39+130=169
14x14=56+140=196
15x15=75+150=225
16x16=96+160=256
17x17=119+170=289 (This is a little harder)
18x18=144+180=324
19x19=171+190=361

Now when we multiply the units digit by a 2 digit number  and the resultant number has three digits it bcomes  a little harder.  This is because we typically don't learn these numbers in our math classes. However, don't let this stop you. You have 3 numbers to memorize right. Not a 100 like you did when you memorized your times tables.

Now lets say that we have 12 x 18, we do this in the same fashion but it is not a square. This looks like:

96 + 120=216 or how about 36 +180=216. Smaller number that result from the units digit result will be easier to utilize.

If it is hard for you to mulitply a one digit by a 2 digit number with a three digit resultant here are some numbers for you to memorize:

6x17=102, 6x18=108, 6 x 19=114
7x15=105, 7x16=112, 7x17=119, 7x18=126, 7x19=133
8X13=104, 8x14=112, 8x15=120,8x16=128,8x17=136,8x18=144, 8x19=152
9x12=108, 9x13=117, 9x14=126, 9x15=135, 9x16=144, 9x17=153, 9x18=162, 9x19=171

That is not a lot to memorize but I can hear the groans as well as the boos and hisses. So lets just learn a methodology so we don't need to remember. Okay?

The easy part will always be the tens digit. Remember this and forget about the 10's digit for now. Say that you want to multiply 17x19.

Lets look at 7x9=63 and 7 times 10 =70. 70 +63 is 133. Like clockwork we have 133 in our heads and we can easily add this to 190. Our answer would then be 323.

The key is to not juggle too many number in our heads. Before calculators people had to use numbers and more people where able to do basic multiplication in their heads. A person who practices this 30 minutes a day for week should have it mastered.

Prime-al Instinct

How to know if a number is prime.

Knowing if a number is prime  may not exactly be an instinct.   However, it is actually fairly easy to determine if the number is prime.  Many schools teach students to just memorize the prime numbers under 100.  I have observed students being asked on a test if a number like 443 is prime.  Most students aren't able to answer that question.

There are many advanced topics on prime numbers and they are not easy to read and are hard to understand so we will avoid those concepts and teach an easy to understand concept.   We will start with the definition of  prime number which is a number whose factors are only 1 and itself.  So to be a non-prime or composite number, the number will havoc to be divisible by some number.  Because every number can be factored down to their prime numbers, we will divide by prime numbers and see if the number is prime.

We remember divisibility rules and know that a number that ends in a 3 cannot be divided by 2 or 5. We will start with:

443÷3 is not evenly divisible
443÷7 is not evenly divisible

443÷11 is not evenly divisible

443÷13 is not evenly divisible

443÷17 is not evenly divisible

443÷19 is not evenly divisible

443÷23 is not evenly divisible

The number is prime.  You might wonder why I stopped at 23 and did not test all the numbers up to 443.  The reason is that 443 is less than 23².  23²=529.    If the number is composite, then it must consist of two different factors and if the number  has a prime factor higher that 23,  there must be a corresponding  prime factor lower than 23.  This is because 23 times any number larger than 23 would be more than 443.  Therefore if 443 is not prime, there will be a number less than 23 which is its factor and we already tested all prime factors less than 23.

Exercise

Try and see if the following numbers are prime:  299, 137, 441, and 537.

Divide evenly:  To divide evenly means that if you divide one number by another there is no remainder.  We can make this easier by saying that it will not produce a non integer.  4 divided by 3 = 1.33333..... which is not an integer so 4 is not evenly divisible by 3.

Composite number:  A number that is composed of two different integers.  With the exception of 1 and 0, a composite number is an integer that is not a prime number.

Answers:  137 and 537 are prime.

In the Prime of Its Life

The first primes less than 100  and rules to easily find

Here is a common scenario:

You are taking the SAT test and you are asked how many prime numbers there are between 30 and 80.  You know what prime numbers are and you know what prime numbers are but you have limited time and no table to help you.  You decide to skip the question because you do not have enough time to finish it by the end of the test.

This is a very common test question yet it is not trivial because most people have not memorized the prime numbers so here is a quick was to determine the prime numbers. 

1   2   3   4   5   6   7   8   9   10            (4 primes)

11 12 13 14 15 16 17 18 19 20            (4 primes)

21 22 23 24 25 26 27 28 29 30            (2 primes)

31 32 33 34 35 36 37 38 39 40            (2 primes)

41 42 43 44 45 46 47 48 49 50            (3 primes)

51 52 53 54 55 56 57 58 59 60            (2 primes)

61 62 63 64 65 66 67 68 69 70            (2 primes)

71 72 73 74 75 76 77 78 79 80            (3 primes)

81 82 83 84 85 86 87 88 89 90            (2 primes)         

91 92 93 94 95 96 97 98 99 100          (1 prime)

                                                 
 The sieve of Eratosthenesis is a methodology for finding all prime numbers.  Without going into how the sieve of Eratosthenes works, I will say the methodology I present will be similar but not the same and it will be easier to use for numbers between 1 and 100.  

After 2, all even numbers are not prime.  
After 5, all numbers that end in 5 are not prime.
No numbers that end in 0 are prime.

Now here is the trick.  If you are looking at the columns where the numbers end in a 1, 3, 7, or 9, you will see there are quite a few primes.  Of these numbers that end in 1, 3, 7, and 9, all numbers except 1, 3, 49, 77, 91, are divisible by 3.

Okay lets check this out.
21÷3=21              33÷3=11               27÷3=9               39÷3=13
51÷3=14              63÷3=21              57÷3=19              81÷3=27
81÷3=27              93÷3=31              87÷3=29              69÷3=23 

You can easily find any prime on numbers less than 100.  If you are asked if 87 is prime, you will say no it isn't because it is divisible by 3.  If you use the 3 divisibility rule, it is even easier.  

Also one more trick for those taking the SAT or any standardized test, the number of primes in each row after the tens are:
 2, 2, 3, 2, 2, 3, 2, 2, 1 and the ones and 10's each have 4.  Refer to the table above.
If you are asked the question, "How many prime numbers are there between 30 and 80?"  You can answer 16.  2+3+2+2+3+2+2=16


                                                              

3 Times is a charm

3 divisibility rule

If a number  is divisible by 3, then the sum of the digits of the number are also divisible by 3.

As an example:

63, 123, 2223 are all divisible by 3.  We can easily determine this in our heads without a calculator by adding the digits and then dividing by 3 as follows:

6+3=9 

9÷3=3

63 is evenly divisible by 3.

If we use a calculator, we find 63÷3=21

1+2+3=6

6÷3=2

123 is divisible by 3.

If we use our a calculator, we find 123÷3=41

2+2+2+3=9

9÷3=3

2223 is divisible by 3
If we use our calculator, we find 2223÷3=741

89, 173, and 3113 are not divisible by 3.  Here is why:

8+9=17 and 17 is not divisible by 3

1+7+3=11 and 11 is not divisible by 3

3+1+1+3=8 and 8 is not divisible by 3

This is a very simple rule yet very powerful.  There are other rules that I will present a little later but this is one of the most important rules and this will help you through many tests like the SAT and ACT.

Try these on your own:

Are these numbers divisible by 3?  Add the digits and divide.  Check with your calculator.

1234, 174, 12761, 25437

Hint all are except 12761.

Prime Time 

What are prime numbers?

A prime number is a positive integer that can only be divided evenly by by 2 different integers 1 and itself.  5 can only be divided evenly by 5 and 1 so this number is prime.  6 can be divided evenly by 2, 3, 6, and 1 so 6 would not be a prime number.

Some fun facts:

• 1 is not a prime number.  It is defined as not prime and is not divisible by 2 different integers.  It is only divisible by 1.
• 2 is the only even prime number.
• Numbers that are greater than one and not prime are called composite numbers.
• There are infinitely many prime numbers.
• 0 is not a prime number.

How do we determine if a number is prime?

You can easily figure out if a number is prime by dividing by smaller prime numbers to see if it can be evenly divided.  If I want to find out if 11 prime, I can try to divide it by smaller prime numbers like 2,3,5, or 7.  11 cannot be evenly divided by any smaller prime factors therefore it is prime.  If I wanted to find out if 33 is prime, I can divide 33 by 3 and I find it will evenly divide into 33 with factors of 1,3,11, and 33, therefore it is not prime.

Following are the prime numbers less than 100:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97

Practice:

Find out if the following numbers are prime:  101, 123 149, and 171.

Definitions:

Divide evenly:  To divide evenly means that if you divide one number by another there is no remainder.  We can make this easier by saying that it will not produce a non integer.  4 divided by 3 = 1.33333..... which is not an integer so 4 is not evenly divisible by 3.