Today’s Puzzle:
It’s early in 2024, so here’s a Factor Fits puzzle utilizing the factors of 20 and 24. Give it a try! There is only one solution.
Factors of 1772:
This is my 1772nd post. What are the factors of 1772?
- 1772 is a composite number.
- Prime factorization: 1772 = 2 × 2 × 443, which can be written 1772 = 2² × 443.
- 1772 has at least one exponent greater than 1 in its prime factorization so √1772 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1772 = (√4)(√443) = 2√443.
- The exponents in the prime factorization are 2 and 1. Adding one to each exponent and multiplying we get (2 + 1)(1 + 1) = 3 × 2 = 6. Therefore 1772 has exactly 6 factors.
- The factors of 1772 are outlined with their factor pair partners in the graphic below.
1772 is a Centered Heptagonal Number:
1772 is one more than 7 times the 22nd triangular number. For all previous centered heptagonal numbers about which I’ve written, I only mentioned their inclusion in this set of numbers. This time, I was determined to produce a graphic of the number. I used Desmos and Excel to determine all 1772 points in the graphic. It was a little time-consuming, but I got it done!
The points of the first heptagon were (1, 0), (cos2π/7, sin2π/7), (cos4π/7, sin4π/7), (cos6π/7, sin6π/7), (cos8π/7, sin8π/7), (cos10π/7, sin10π/7), (cos12π/7, sin12π/7). Here is an example of what was involved in completing one side of the other heptagons: Suppose I wanted to find five points on the line connecting (a,c) and (b,d). The five points would be
((4a+0b)/4, (4c+0d)/4), or simply (a, c),
((3a+1b)/4, (3c+1d)/4),
((2a+2b)/4, (2c+2d)/4), or simply ((a+b)/2, (c+d)/2), the midpoint,
((1a+3b)/4, (1c+3d)/4),
((0a+4b)/4, (0c+4d)/4), or simply (b, d).
I used Excel to calculate those numbers and then copied and pasted them into Desmos which graphed them beautifully. Each round took me less than ten minutes to complete. Here is the finished product:
More About the Number 1772:
1772 is the difference of two squares:
444² – 442² = 1772.
1772 is palindrome 24042 in base 5. Why?
Because 2(5⁴)+4(5³)+0(5²)+4(5¹)+2(5º) = 1772.