Awesome Numbers

{Awesome Works in Progress}

• 6174 is known as Kaprekar's constant
• 1089 - 1089 is widely used in magic tricks because it can be "produced" from any two three-digit numbers. This allows it to be used as the basis for a Magician's Choice.
• 369 (Key of the Universe) - Unlocking the Secrets of 369: Understanding Tesla’s Theory of Numbers
• 1729 is known as the Hardy-Ramanujan number, and it is the smallest number that can be expressed as the sum of two cubes in two different ways: 1729 = 1^3 + 12^3 = 9^3 + 10^3.
• 2187: This number is known as the 3-cube, and it is the smallest integer that can be expressed as the cube of an integer in three different ways: 2187 = 3^3 = 6^3 = 9^3.
• 4104: This number is known as the Brocard number, and it is the smallest integer that can be expressed as the sum of two squares in two different ways: 4104 = 66^2 + 88^2 = 94^2 + 32^2.
• 5040: This number is known as the Euler number, and it is the smallest integer that can be expressed as the sum of two squares in three different ways: 5040 = 70^2 + 54^2 = 72^2 + 30^2 = 80^2 + 24^2.
• 720720: This number is known as the Schutz number, and it is the smallest integer that can be expressed as the sum of two squares in four different ways: 720720 = 880^2 + 464^2 = 936^2 + 168^2 = 968^2 + 120^2 = 1008^2 + 64^2.
• 8128: This number is known as the Sophie Germain prime, and it is the smallest number that can be expressed as the sum of two squares in five different ways: 8128 = 90^2 + 60^2 = 96^2 + 48^2 = 98^2 + 42^2 = 102^2 + 36^2 = 108^2 + 24^2.
• 10648: This number is known as the Taxicab number, and it is the smallest integer that can be expressed as the sum of two cubes in three different ways: 10648 = 104^3 + 24^3 = 110^3 + 9^3 = 121^3 + 1^3.
• 20736: This number is known as the Ramanujan number, and it is the smallest integer that can be expressed as the sum of two cubes in four different ways: 20736 = 162^3 + 36^3 = 170^3 + 15^3 = 180^3 + 6^3 = 192^3 + 1^3.
• 2520: This number is known as the least common multiple of the first ten positive integers, and it is the smallest integer that is divisible by all of the integers from 1 to 10.
• 2821109907456: This number is known as the Moser number, and it is the smallest integer that can be expressed as the sum of two cubes in five different ways: 2821109907456 = 2342^3 + 746^3 = 2420^3 + 532^3 = 2542^3 + 300^3 = 2772^3 + 32^3 = 3062^3 + 6^3.
• 4, 2, 1 loop

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• π 3.14159 What Is PI?
• 2.71828 Euler's Number (e) What's so special about Euler's number e?

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Rules

• The Rule of 72
• Any natural number greater than 6 that ends with a 6 is always composite Learn more
• Blue-Seven Phenomenon 📺 ~24min - Veritasium

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Divisibility Rules

• Divisibility by 2: A number is divisible by 2 if its last digit is even (0, 2, 4, 6, or 8).
• Divisibility by 3: A number is divisible by 3 if the sum of its digits is divisible by 3.
• Divisibility by 4: A number is divisible by 4 if the number formed by its last two digits is divisible by 4.
• Divisibility by 5: A number is divisible by 5 if its last digit is 0 or 5.
• Divisibility by 6: A number is divisible by 6 if it is divisible by both 2 and 3 (i.e., it's even and the sum of its digits is divisible by 3).
• Divisibility by 7: This one is a bit more complex. Double the last digit, subtract it from the rest of the number, and if the result is divisible by 7 (or is 0), then the original number is divisible by 7. This process can be repeated if necessary.
  • Using the number 161 as an example for the divisibility by 7 rule:
  • Double the last digit: The last digit is 1, so doubling it gives us 2.
  • Subtract this from the rest of the number: The rest of the number, excluding the last digit, is 16. So, 16 - 2 = 14.
  • Check if the result is divisible by 7: 14 is divisible by 7.
• Divisibility by 8: A number is divisible by 8 if the number formed by its last three digits is divisible by 8.
• Divisibility by 9: If the sum of the digits of a number is divisible by 9, then the number itself is divisible by 9. This rule is a useful shortcut to determine divisibility by 9. For example, if you have a number like 162, the sum of its digits is 1 + 6 + 2 = 9. Since 9 is divisible by 9, the number 162 is also divisible by 9.
• Divisibility by 10: A number is divisible by 10 if its last digit is 0.