# 1098 a Lucky Level 4 Puzzle?

You are lucky that this puzzle has some easy clues in it. You will have no problem getting started. Be warned, later on, you may not feel so lucky! I’m sure you can solve it if you keep with it. Print the puzzles or type the solution in this excel file: 12 factors 1095-1101

Now, here are a few facts about the number 1098:

• 1098 is a composite number.
• Prime factorization: 1098 = 2 × 3 × 3 × 61, which can be written 1098 = 2 × 3² × 61
• The exponents in the prime factorization are 1, 2, and 1. Adding one to each and multiplying we get (1 + 1)(2 + 1)(1 + 1) = 2 × 3 × 2 = 12. Therefore 1098 has exactly 12 factors.
• Factors of 1098: 1, 2, 3, 6, 9, 18, 61, 122, 183, 366, 549, 1098
• Factor pairs: 1098 = 1 × 1098, 2 × 549, 3 × 366, 6 × 183, 9 × 122, or 18 × 61,
• Taking the factor pair with the largest square number factor, we get √1098 = (√9)(√122) = 3√122 ≈ 33.13608 This first fact from OEIS.org uses only digits found in 1098. It makes 1098 look pretty lucky:
1098 = 11 + 0 + 999 + 88

1098 is the sum of four consecutive prime numbers:
269 + 271 + 277 + 281 = 1098

1098 is the hypotenuse of a Pythagorean triple:
198-1080-1098 which is 18 times (11-60-61)

1098 is a palindrome when it is written in three different bases:
It’s 2112 in BASE 8 because 2(8³) + 8² + 8 + 2(1) = 1098,
909 in BASE 11 because 9(11² + 1) = 9(122) = 1098, and it’s
666 in BASE 13. Oh my! How unlucky can you get? Why does it have two unlucky numbers, 666 and 13? Because 6(13² + 13 + 1) = 6(183) = 1098.

# 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