# 899 and Level 3

Yesterday I worked with a student who knew that 5 × 5 = 25 but couldn’t remember what 4 × 6 is. I didn’t use the variable “n”, but I used some examples from the times table to help her understand that (n-1)(n+1) = n² – 1. Always. Then I said to her, “30 × 30 = 900, so how much is 29 × 31?” She answered, “899?” I told her, “Yes, it is.” She was pretty pleased with herself.

29 and 31 are twin primes, so that makes 29 × 31 = 899 even cooler.

899 is also the hypotenuse of a Pythagorean triple:

• 620-651-899 which is (20-21-29) times 31.

Print the puzzles or type the solution on this excel file: 10-factors-897-904

• 899 is a composite number.
• Prime factorization: 899 = 29 × 31
• The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 899 has exactly 4 factors.
• Factors of 899: 1, 29, 31, 899
• Factor pairs: 899 = 1 × 899 or 29 × 31
• 899 has no square factors that allow its square root to be simplified. √899 ≈ 29.9833