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What is the sum of all of the angles that are shaded green?

The number 37 is a very interesting number. It is a prime number so it is divisible by 1 and itself. But it is the only prime number which can do the following

111 ÷ 37 = 3 222 ÷ 37 = 6 333 ÷ 37 = 9 444 ÷ 37 = 12 555 ÷ 37 = 15 666 ÷ 37 = 18

What will you get when 777, 888 and 999 are divided by 37 ?

Unique PrimeThe number 142857 is rather special. When you multiply it the digits found in the answer stay the same, just in a different order. Check it out:

1 x 142857 = 142857 2 x 142857 = 285714 3 x 142857 = 428571 4 x 142857 = 571428 5 x 142857 = 714285 6 x 142857 = 857142

What happens when you multiply it by 7?

India's Discovery

The concept of zero was developed in India as early as the Gupta period in the 5th century. The Indian scholar Pingala (c. 200 BC) used the Sanskrit word sunya explicitly to refer to zero. The origin of the modern decimal-based place value notation can be traced to the Aryabhatiya(c. 500), which states sthānāt sthānaṁ daśaguṇaṁ syāt " from place to place each is ten times the preceding".

Pi is the most famous irrational number, meaning that its decimal form neither ends (like 1/4 = 0.25) nor becomes repetitive (like 1/6 = 0.166666…). (To only 18 decimal places, pi is 3.141592653589793238). March 14 is celebrated as the international Pi day. Many competitions are held where contestants recite pi to biggest decimal places they can remember. How many decimal places can you recite ?

Famous Irrational Number

Prime numbers are positive integers that are only divisible by itself and 1. The numbers 2,3,5,7…are all prime. The great Greek mathematician Euclid had proven that there are an infinite numbers of prime numbers as early as 300 B.C. Did you know that in the banking system most of the encryption are done using prime numbers. The largest prime number found so far is 2^{57,885,161}-1 which has 17,425,170 digits.

Amicable numbers love each other so much. How much? Well, lets take a classic pair 284 and 220 and see just how friendly they are. Lets take all the proper divisors of 220 (that is to say, all its divisors that leave no remainder, including the number 1, and excluding the number itself) and add them all up: 1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 = 284. Now, lets take 220 and do the same thing: 1 + 2 + 4 +71 + 142 = 220. These are a pair of amicable numbers. Other pairs include (1184, 1210), (2620, 2924), and (5020, 5564). This type of number pair was discovered and studied by the Pythagoreans and has been the subject of much research through the centuries.

**"Emirp"** is the word "prime" spelled backwards, and it refers to a prime number that becomes a new prime number when you reverse its digits. Emirps do not include palindromic primes (like 151 or 787), nor 1-digit primes like 7. The first few emirps are 13, 17, 31, 37, 71, 73, 79, 97, 107, 113, 149, and 157 – reverse them and you have got a new prime number on your hands.

Emirp
Prime

Abundant numbers, also known as "excessive," are bigger than the sum of their proper divisors. 12, for instance, is the first (smallest) abundant number–the sum of its proper divisors, 1+2+3+4+6, is 16. 12, therefore, has an "abundance" of 4, the amount by which the sum of its divisors exceeds the number.

There are many even abundant numbers, but we dont get to an odd one until the number 945. Some abundant numbers are "semiperfect" or "pseudoperfect," meaning that they are equal to all or just some of their proper divisors. An example of imperfect number is 12.

Achilles was a powerful Trojan War hero who was extremely powerful but had one flaw his achilles heel. Like him, **Achilles numbers** are powerful but not perfect. So, lets begin with a powerful number. A number is considered powerful if all of its prime factors remain factors once they are squared. 25 is a powerful number because its one prime factor, 5, remains a factor once its been squared (25, which goes into 25 once) Now lets move onto perfect powers, number that can be expressed as an integer power of another integer;

8 is a perfect power, as its 2 cubed.So now, back to the original premise Achilles numbers are** powerful**, but they are not** perfect powers**. 72 is the first Achilles number, as it is powerful, but it is not a perfect prime. Others include 108, 200, 288, 392, 432, 500, and 648.

Some numbers are weird; others are **happy**. If you would like to find out if a given number is happy, you will need to perform the following set of operations. Lets take the number 44:

First, square each digit, then add them together: 4^{2} + 4^{2} = 16 + 16 = 32

Then, we will do it again with our new number: 3^{2} + 2^{2} = 9 + 4 = 13

And again:
1^{2} + 3^{2} = 1 + 9 = 10 And finally: 1^{2} + 0^{2} = 1 + 0 = 1

It's a happy number! Anytime you take a number, perform this "procedure," and eventually arrive at the number 1, you have yourself a happy number. If your number never reaches 1, then sadly, its unhappy. Interestingly, happy number are extremely common; there are 11 of them between 1 and 50, for example.
As a final note, the greatest happy number with no recurring digits is 986,543,210. That is a happy number indeed.

**Narcissistic numbers**, also known as Armstrong numbers or "pluperfect digital invariants," are numbers that listen closely are equal to the sum of each of its digits when those digits are raised to the power of the AMOUNT of digits in the number. Ok. What? Lets take an example of the four existing narcissistic cubes:

153 = 1^{3} + 5^{3} + 3^{3}
370 = 3^{3} + 7^{3} + 0^{3}
371 = 3^{3} + 7^{3} + 1^{3}
407 = 4^{3} + 0^{3} + 7^{3}
In these cases, each digit is cubed because there are three digits in the number. Then, those cubed numbers are added together to produce a sum equal to the original number. There are no 1 digit narcissistic numbers, nor 12 or 13 digit ones; the two 39 digit ones are: 115132219018763992565095597973971522400 and 115132219018763992565095597973971522401.

The number 37 is a very interesting number. It is a prime number so it is divisible by 1 and itself. But it is the only prime number which can do the following

111 ÷ 37 = 3 222 ÷ 37 = 6 333 ÷ 37 = 9 444 ÷ 37 = 12 555 ÷ 37 = 15 666 ÷ 37 = 18

What will you get when 777, 888 and 999 are divided by 37 ?

Unique PrimeThe number 142857 is rather special. When you multiply by it, the digits found in the answer stay the same, just in a different order. Check it out:

1 x 142857 = 142857 2 x 142857 = 285714 3 x 142857 = 428571 4 x 142857 = 571428 5 x 142857 = 714285 6 x 142857 = 857142

What happens when you multiply it by 7?

India's Discovery

The concept of zero was developed in India as early as the Gupta period in the 5th century. The Indian scholar Pingala (c. 200 BC) used the Sanskrit word sunya explicitly to refer to zero. The origin of the modern decimal-based place value notation can be traced to the Aryabhatiya(c. 500), which states sthānāt sthānaṁ daśaguṇaṁ syāt "from place to place each is ten times the preceding".

Pi is the most famous irrational number, meaning that its decimal form neither ends (like 1/4 = 0.25) nor becomes repetitive (like 1/6 = 0.166666…).

(To only 18 decimal places, pi is 3.141592653589793238). March 14 is celebrated as the international Pi day. Many competitions are held where contestants recite pi to biggest decimal places they can remember. How many decimal places can you recite ?

^{57,885,161}-1 which has 17,425,170 digits.

Amicable numbers love each other so much. How much? Well, lets take a classic pair 284 and 220 and see just how friendly they are. Lets take all the proper divisors of 220 (that is to say, all its divisors that leave no remainder, including the number 1, and excluding the number itself) and add them all up: 1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 = 284. Now, lets take 220 and do the same thing: 1 + 2 + 4 +71 + 142 = 220. These are a pair of amicable numbers. Other pairs include (1184, 1210), (2620, 2924), and (5020, 5564).

**"Emirp"** is the word "prime" spelled backwards, and it refers to a prime number that becomes a new prime number when you reverse its digits. Emirps do not include palindromic primes (like 151 or 787), nor 1-digit primes like 7. The first few emirps are 13, 17, 31, 37, 71, 73, 79, 97, 107, 113, 149, and 157 – reverse them and you have got a new prime number on your hands.

Abundant numbers, also known as "excessive," are bigger than the sum of their proper divisors. 12, for instance, is the first (smallest) abundant number–the sum of its proper divisors, 1+2+3+4+6, is 16. 12, therefore, has an "abundance" of 4, the amount by which the sum of its divisors exceeds the number.

There are many even abundant numbers, but we dont get to an odd one until the number 945. Some abundant numbers are "semiperfect" or "pseudoperfect," meaning that they are equal to all or just some of their proper divisors. An example of imperfect number is 12.

Achilles was a powerful Trojan War hero who was extremely powerful but had one flaw his achilles heel. Like him, **Achilles numbers** are powerful but not perfect. A number is considered powerful if all of its prime factors remain factors once they are squared. 25 is a powerful number because its one prime factor, 5, remains a factor once its been squared (25, which goes into 25 once);

8 is a perfect power, as its 2 cubed.72 is the first Achilles number, as it is powerful, but it is not a perfect prime.

Achilles NumberSome numbers are weird; others are **happy**. If you would like to find out if a given number is happy, you will need to perform the following set of operations. Lets take the number 44:

First, square each digit, then add them together: 4^{2} + 4^{2} = 16 + 16 = 32 Then, we will do it again with our new number: 3^{2} + 2^{2} = 9 + 4 = 13 And again:
1^{2} + 3^{2} = 1 + 9 = 10 And finally: 1^{2} + 0^{2} = 1 + 0 = 1 Anytime you take a number, perform this "procedure," and eventually arrive at the number 1, you have a happy number. Interestingly, happy number are extremely common; there are 11 of them between 1 and 50.
The greatest happy number with no recurring digits is 986,543,210.

**Narcissistic numbers**, also known as Armstrong numbers are numbers that are equal to the sum of each of its digits when those digits are raised to the power of the AMOUNT of digits in the number. Ok. What? Lets take an example of the two existing narcissistic cubes:

153 = 1^{3} + 5^{3} + 3^{3}
370 = 3^{3} + 7^{3} + 0^{3}
In these cases, each digit is cubed because there are three digits in the number. Then, those cubed numbers are added together to produce a sum equal to the original number. There are no 1 digit narcissistic numbers, nor 12 or 13 digit ones.