Properties of Number 8

        Importance of  Eight (8)

1.      It is a composite number with its proper divisor being 1, 2 and 4.

2.      It is the base of the octal number system. In octal number system one digit represents 3 bits.

3.      The atomic number of oxygen is eight.

4.      Scorpio is the 8^th astrological sign of the Zodiac.

5.      A map consists of 8 directions:- North, Northeast, Northwest, East, Southeast, Southwest, West, South.

6.      In three dimensional spaces there are 8 octants and the eight octants have the different co-ordinates shown here.

I	II	III	IV	V	VI	VII	VIII
+	_	_	+	+	_	_	+
+	+	_	_	+	+	_	_
+	+	+	+	_	_	_	_

7.      Apart from the trivial cases of 0 and 1, 8 is the smallest number for which the sum of the digits of its cube is equal to the number.                                                                                    

                    8³         = 512  and 5 + 1 + 2 = 8

8.      It is the largest cube in the Fibonacci sequence. A Fibonacci number is a numbers in which the next number is the sum of previous two. 1, 1, 2, 3, 5, 8… are the numbers of the Fibonacci sequence.

9.      It is the second cube number. 2³ = 8

10.  One can split a pie into 8 pieces with three straight cuts.

11.  A cube or cuboids has 8 vertices.

12.  An umbrella is octagonal in shape.    

13.  The digital sum of a prime pair product is 8.                                                                 

                          5 x 7 = 35 and 3 + 5 = 8                                                              

                          11 x 13 = 143 and 1 + 4 + 3 = 8                                                                               

                          17 x 19 = 323 and 3 + 2 + 3 = 8                                                                                 

                       29 x 31 = 899 and 8 + 9 + 9 = 26 and 2+ 6 = 8

14.  A number is divisible by 8 if it ends with 000 or its last three digit is divisible by 8. 

                          Example – 1000, 456640

15.  Many spiders, bees and flies have eight legs.

16.  Eight is the third number that remains the same written upside down.

17.  The amount of cloud in the sky is measured in oktas on a scale from zero to eight. 8 oktas means the sky is totally clouded.

18.  It is the second honest number.                                                                                                      

                2³ = Two Cubed = 8 has eight letters.

19.  It is the smallest sum of two factorials of distinct primes.                                                              

                       2! + 3! = 2 + 6 = 8

20.  It is the only cube that is 1 less than a square.                                                                                  

                           8 = 3² – 1 

21.  Digit sum of prime pair product comes out 8.                                                                                  

                5 x 7 = 35                           and        3 + 5 = 8                                                                      

              11 x 13 = 143  and                                   1 + 4 + 3 = 8                                                                

              17 x 19 = 323  and      3 + 2 + 3 = 8

22.  Eight can be portioned in 22 ways.

23.  8 times any triangular number is 1 less than a square. 1, 3, 6, 10,… are the examples of triangular number. If we take 3 as a triangular number than 8x 3 = 24 is 1 less than 25 which is a square number.

24.  It is the second octagonal number represented by the formula n(3n – 2 ), where n is any natural number. The sequence is 1, 8, 21, 40….

Importance of Number 7

         Characteristics of Seven (7)

1.  It is a prime number. Prime number is a number which has only two factors 1 and the number itself.

2.  There are seven days in a week. They are – Sunday, Monday, Tuesday, Wednesday, Thursday, Friday and Saturday.

3. The oldest puzzle recorded in Ahmes Papyrus has another version like this—             

           As I was going to St. Ives,                                                                         

            I met a men with 7 wives.                                                                                                                Each wife had 7 sacks.                                                                                                               

        Each sack has 7 cats.                                                                                                        

        Each cat had 7 kits.                                                                                         

      Kits, cats, sack and wives                                                                     

  how many were going to St. Ives?                                                                      

This puzzle requires the knowledge of Geometric Progression. In simple words, the total number = 1 + 7 + 49 + 343 + 2401 = 2801.

4. Seven (7) is the congruent number because it is the area of the triangle with sides 35/12, 24/5 and 337/60. A triangle is said to be congruent if its area comes out to be an integer.

5. It is a Mersenne Prime number. A prime which can be written as 2ⁿ – 1 is said to be Mersenne Prime.                                                                                                                                                            7 = 2³ – 1

6. The sevenths power of number 14459929 has a very interesting feature.  7^th  Power of every digit of 14459929 sums up to 14459929.                                                                           

           14459929 = 1⁷ + 4⁷ + 4⁷ + 5⁷ + 9⁷ + 9⁷ + 2⁷ + 9⁷

7. The factorial of seven (7!) is 5040. This result contains 4 digits with 2 zeros, i.e. 50% zero. Probably it is the unique factorial with this property.

8. No perfect square ends with 7.

9. The smallest positive integer whose reciprocal 1/7 has a pattern of more than one repeating digit.  Example:- ¹/₇ = 0.142857 142857 142857….This number 142857 shows an explicit feature when it is multiplied by 1, 2, 3, 4, 5 and 6. Let’s see it-                                                                    

  142857 x 1 = 142857                                                                                                 

    142857 x 2 = 285714                                                                                                       

   142857 x 3 = 428571                                                                                                        

   142857 x 4 = 571428                                                                                                     

   142857 x 5 = 714285                                                                                                                      142857 x 6 = 857142                                                                           

Did you notice anything? Yes, the digit of the product comes in a cyclic order and when you prepare a multiplicative table on the basis of the result of the above product the sum of each row and column comes out to be 27.

Digit	1	4	2	8	5	7	Digit Sum
1	1	4	2	8	5	7	27
2	2	8	5	7	1	4	27
3	4	2	8	5	7	1	27
4	5	7	1	4	2	8	27
5	7	1	4	2	8	5	27
6	8	5	7	1	4	2	27
Column sum	27	27	27	27	27	27

10. Seven can be portioned in 11 ways.

11. The seven bridges of Konigsberg in Germany gave birth to new branch of Mathematics called Topology. Leonard Euler (1707-1783) solved this problem.

                                        

12. No perfect square ends with 7. Any number that ends with 2, 3, 7 or 8 can’t be a perfect square.

13. It is a Lucas Number. 1, 3, 4, 7, 11, 18… are the Lucas number. In Lucas Number, the next number is the sum of previous two.

14. It is the sum of 1^st and 3^rd factorials. 1! + 3!  = 7

15. It is the first centered hexagonal number.

                                       

1.      There are 7 different ways of linking four hexagons together. One of them is shown here.

                

2.       The opposite side of a die cube sum up to 7.

1.      It is the first prime which can be written in the form of 6m + 1, where m is a positive integers.

2.       Here is an interesting game based on 7.                                                                                 

               7 is a Prime Number                                                                              

             73 is a Prime Number                                                                                               

            739 is a Prime Number.                                                                                            

           7393 is a Prime Number.                                                                                           

         73939 is a Prime Number.                                                                                                     

        739391 is a Prime Number.                                                                                                   

      7393913 is a Prime Number.

3.      In the Fermat’s Last Theorem xⁿ + yⁿ = zⁿ, for n = 7 this equation has no solution was proved by Lame in 1840.

14. The sum of the squares of the first seven primes was 666, the supposed number of the beast from Revelations.
666=2^2+3^2+5^2+7^2+11^2+13^2+17^2

15. The smallest prime Palindrome number 7778777 has only two digits in it ,i.e 7 and 8.

  

     Send your comments to 

        Rajesh Kumar Thakur

      rkthakur1974@gmail.com