# How Lucky Can 777 Be?

• 777 is a composite number.
• Prime factorization: 777 = 3 x 7 x 37
• The exponents in the prime factorization are 1, 1, and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 x 2 x 2 = 8. Therefore 777 has exactly 8 factors.
• Factors of 777: 1, 3, 7, 21, 37, 111, 259, 777
• Factor pairs: 777 = 1 x 777, 3 x 259, 7 x 111, or 21 x 37
• 777 has no square factors that allow its square root to be simplified. 77727.8747197. Some people think that 7 is a lucky number. If that is true, then 777 should be even luckier.

Some numbers are lucky enough to be included in Multiplication Rhymes.ppt – mathval, a fun power point that helps students learn 12 multiplication facts, including these three that use Lucky Numbers:

• 3 & 7 are always lucky numbers; 3 x 3 = 9 lives of a cat.
• 3 & 7 are always lucky numbers; 3 x 7 = 21 lucky age.
• 3 & 7 are always lucky numbers; 7 x 7 = 49er Gold Miner.

In Number Theory Lucky Numbers are actually defined and can be generated using a sieve somewhat similar to the prime number generating Sieve of Eratosthenes. There is an infinite number of Lucky Numbers, and yes, 3, 7, 9, 21, 49, and 777 all make the list.

Oeis.org’s wiki, Lucky numbers, includes several lucky number lists including the first 33 composite Lucky Numbers thus defined because ALL of their factors are also Lucky Numbers. 777 was the 19th number on that particular list because ALL of its factors, 1, 3, 7, 21, 37, 111, 259, 777 are Lucky Numbers, too!

If that isn’t lucky enough, 777 is a repdigit in three different bases.

• 3333 BASE 6; note that 3(6^3) + 3(6^2) + 3(6^1) + 3(6^0) = 777
• 777 BASE 10; note that 7(100) + 7(10) + 7(1) = 777
• LL BASE 36 (L is 21 base 10); note that 21(36) + 21(1) = 777

Did you notice that lucky numbers 3, 7, and 21 showed  up again? I liked that coincidence so much that I made this graphic: 777 is also the sum of three squares four different ways:

• 26² + 10² + 1² = 777
• 22² + 17² + 2² = 777
• 20² + 19² + 4² = 777
• 20² + 16² + 11² = 777

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