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.

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