# 837 and Level 3

837 is repdigit RR in BASE 30 (R is 27 in base 10). All that means is that 27(30¹) + 27(30º) = 837, which naturally follows from the fact that 27(30 + 1) = 837.

837 has four odd factor pairs, so 837 can be written as the difference of two squares four different ways:

• 837 × 1 = 837 means 419² – 418² = 837
• 279 × 3 = 837 means 141² – 138² = 837
• 93 × 9 = 837 means 51² – 42² = 837
• 31 × 27 = 837 means 29² – 2² = 837

Hmm…837 is only four numbers away from the next perfect square, 841. Print the puzzles or type the solution on this excel file: 10-factors-835-842

• 837 is a composite number.
• Prime factorization: 837 = 3 × 3 × 3 × 31, which can be written 837 = 3³ × 31
• The exponents in the prime factorization are 3 and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1) = 4 × 2 = 8. Therefore 837 has exactly 8 factors.
• Factors of 837: 1, 3, 9, 27, 31, 93, 279, 837
• Factor pairs: 837 = 1 × 837, 3 × 279, 9 × 93, or 27 × 31
• Taking the factor pair with the largest square number factor, we get √837 = (√9)(√93) = 3√93 ≈ 28.93095  This site uses Akismet to reduce spam. Learn how your comment data is processed.