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.

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