Today’s Puzzle:

The gift of a single red rose is a way to say, “I love you.” To me, a single red rosebud would be saying, “I love you, and my love for you is growing.” To all my faithful readers, I give you this single red rosebud:

Here is the same puzzle without any added color if you want to save on printer ink.

Factors of 1776:

Another way to show love is to plant a tree. How about we plant a factor tree? Since 1776 has twenty different factor pairs, MANY possible factor trees could be planted. I chose to base this one on the fun fact that 1776 = 4 · 444:

• 1776 is a composite number.
• Prime factorization: 1776 = 2 × 2 × 2 × 2 × 3 × 37, which can be written 1776 = 2⁴ × 3 × 37
• 1776 has at least one exponent greater than 1 in its prime factorization so √1776 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1776 = (√16)(√111) = 4√111
• The exponents in the prime factorization are 4, 1 and 1. Adding one to each exponent and multiplying we get (4 + 1)(1 + 1)(1 + 1) = 5 × 2 × 2 = 20. Therefore 1776 has exactly 20 factors.
• The factors of 1776 are outlined with their factor pair partners in the graphic below.

Did you notice all the repdigit factors of 1776 in the table?

More About the Number 1776:

1776 is the hypotenuse of a Pythagorean triple:
576-1680-1776, which is (12-35-37) times 48.

1776 looks interesting in some other bases:
It’s 5115 in base7 because 5(7³) + 1(7²) +1(7) + 5(1) = 1776, and uh oh!
OO in base73 because 24(73) + 24(1) = 24(74) = 1776.
We’ve run out of letters in the alphabet to use as numbers, but I
will note that 37(47) + 37(1) = 37(48) = 1776.