The theory of numbers is particularly liable to the accusation that some of its problems are the wrong sort of questions to ask. I do not myself think the danger is serious; either a reasonable amount of concentration leads to new ideas or methods of obvious interest, or else one just leaves the problem alone. “Perfect numbers” certainly never did any good, but then they never did any particular harm.
John Edensor Littlewood
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Mathematics is all about numbers and in this blogpost, let us play with them. Actually, we will consider few numbers and discuss about their peculiarity. Let us start with the number 6.
The number 5 is a very special number. The speciality is shown below.
| 5 × 1 = 5 | = | 5 | |
| 5 × 2 = 10 | = | (1 + 0) = 1 | |
| 5 × 3 = 15 | = | (1 + 5) = 6 | |
| 5 × 4 = 20 | = | (2 + 0) = 2 | |
| 5 × 5 = 25 | = | (2 + 5) = 7 | |
| 5 × 6 = 30 | = | (3 + 0) = 3 | |
| 5 × 7 = 35 | = | (3 + 5) = 8 | |
| 5 × 8 = 40 | = | (4 + 0) = 4 | |
| 5 × 9 = 45 | = | (4 + 5) = 9 | |
| 5 × 10 = 50 | = | (5 + 0) = 5 | |
| 5 × 11 = 55 | = | (5 + 5) = 10 and (1 + 0) = 1 | |
| 5 × 12 = 60 | = | (6 + 0) = 6 | |
| 5 × 13 = 65 | = | (6 + 5) = 11 and (1 + 1) = 2 | |
| 5 × 14 = 70 | = | (7 + 0) = 7 | |
| 5 × 15 = 75 | = | (7 + 5) = 12 and (1 + 2) = 3 | |
| 5 × 16 = 80 | = | (8 + 0) = 8 | |
| 5 × 17 = 85 | = | (8 + 5) = 13 and (1 + 3) = 4 | |
| 5 × 18 = 90 | = | (9 + 0) = 9 | |
| 5 × 19 = 95 | = | 9 + 5 = 14 and (1 + 4) = 5 | |
| 5 × 20 = 100 | = | (1 + 0 + 0) = 1 | |
| 5 × 21 = 105 | = | (1 + 0 + 5) = 6 | |
| … … … | … … … |
The pattern 1, 2, 3, 4, 5, 6, 7, 8, 9, . . . keeps on repeating in various order.
The number 6 is a very special number. The speciality is shown below.
| 6 × 1 = 6 | = | 6 |
| 6 × 2 = 12 | = | (1 + 2) = 3 |
| 6 × 3 = 18 | = | (1 + 8) = 9 |
| 6 × 4 = 24 | = | (2 + 4) = 6 |
| 6 × 5 = 30 | = | (3 + 0) = 3 |
| 6 × 6 = 36 | = | (3 + 6) = 9 |
| 6 × 7 = 42 | = | (4 + 2) = 6 |
| 6 × 8 = 48 | = | (4 + 8) = 12 and (1 + 2) = 3 |
| 6 × 9 = 54 | = | (5 + 4) = 9 |
| … … | … … |
The pattern 6, 3, 9, 6, . . . keeps on repeating.
The number 8 is a very special number. The speciality is shown below.
| 8 × 1 = 8 | = | 8 |
| 8 × 2 = 16 | = | (1 + 6) = 7 |
| 8 × 3 = 24 | = | (2 + 4) = 6 |
| 8 × 4 = 32 | = | (3 + 2) = 5 |
| 8 × 5 = 40 | = | (4 + 0) = 4 |
| 8 × 6 = 48 | = | (4 + 8) = 12 and (1 + 2) = 3 |
| 8 × 7 = 56 | = | (5 + 6) = 11 and (1 + 1) = 2 |
| 8 × 8 = 64 | = | (6 + 4) = 10 and (1 + 0) = 1 |
| 8 × 9 = 72 | = | (7 + 2) = 9 |
| 8 × 10 = 80 | = | (8 + 0) = 8 |
| 8 × 11 = 88 | = | (8 + 8) = 16 and (1 + 6) = 7 |
| … … | … … … |
We get a decreasing order pattern 8, 7, 6, 5, 4, 3, 2, 1, 9, 8, . . .
It is the smallest even number, divisible by all the numbers from 1 to 10. We can write 2520 as 7 × 30 × 12. It’s like
7 (number of days in a week) × 30 (number of days in a month) × 12 (number of months in a year).
Split the number 2520 as 25 and 20 and take their sum. The sum will be 25 + 20 = 45. Square of 45 is 452 = 2025 and it’s just the interchange of the numbers 25 and 20!
The number 22520 is a special number, because it is the smallest positive number (> 1), which can be expressed as a first power, second power, third power, fourth power, fifth power, sixth power, seventh power, eight power, and a ninth power. See below.
(21260)2 (2840)3 (2630)4 (2504)5
(2420)6 (2360)7 (2315)8 (2280)9.
73939133 is an interesting number as shown below
- 7 is prime
- 73 is prime
- 739 is prime
- 7393 is prime
- 73939 is prime
- 739391 is prime
- 7393913 is prime
- 73939133 is prime
It’s the time for 381654729 to discuss about its speciality. See below.
- First of all, it contains all the digits from 1 to 9 exactly once.
- The first two digits (from the left) form a number that is divisible by 2;
- The first three digits (from the left) form a number that is divisible by 3;
- The first four digits (from the left) form a number that is divisible by 4;
- The first five digits (from the left) form a number that is divisible by 5;
- The first six digits (from the left) form a number that is divisible by 6;
- The first seven digits (from the left) form a number that is divisible by 7;
- The first eight digits (from the left) form a number that is divisible by 8;
- The first nine digits (from the left) form a number that is divisible by 9;
Your suggestions are eagerly and respectfully welcome! See you soon with a new mathematics blog that you and I call “Math1089 – Mathematics for All!“.