Today’s Puzzle:

Today is Monday, February 19. In the United States, we are celebrating Presidents’ Day, honoring most especially two important presidents who were born in February.

Exactly one week ago was February 12.

George Washington was born on February 11, 1731, Julian calendar.
Abraham Lincoln was born on February 12, 1809, Gregorian calendar.

The Julian calendar didn’t have leap days, so in 1752 a year and eleven days were added to Washington’s birthday to convert it to the Gregorian calendar.

Neither president will ever have his birthday on the third Monday of February when Presidents’ Day is observed. Too bad the second Monday of February wasn’t chosen instead. Then we could fudge a little and say that Presidents’ Day would be observed on one of their birthdays 2/7 of the time!

What days of the month are the earliest and the latest that a second Monday could be? 

When I was young I was told the story about George Washington chopping down a cherry tree. When he was confronted, he would not and could not tell a lie, and confessed his misdeed. As I got older, I learned that this was a fabricated story designed to teach children honesty of all things!

Nevertheless, some people celebrate Presidents’ Day by eating a cherry pie in remembrance of that story.

Factors of 1782:

This is my 1782nd post. Since it’s Presidents’ Day, I thought I would make a few factor trees for that number. You could think of the prime factors in red as cherries on the trees. Notice that all the prime factors are low-hanging fruit on these particular trees!

• 1782 is a composite number.
• Prime factorization: 1782 = 2 × 3 × 3 × 3 × 3 × 11, which can be written 1782 = 2 × 3⁴ × 11.
• 1782 has at least one exponent greater than 1 in its prime factorization so √1782 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1782 = (√22)(√81) = 9√22. The exponents in the prime factorization are 1, 4, and 1. Adding one to each exponent and multiplying we get (1 + 1)(4 + 1)(1 + 1) = 2 × 5 × 2 = 20. Therefore 1782 has exactly 20 factors.
• The factors of 1782 are outlined with their factor pair partners in the graphic below.

More About the Number 1782:

(5(27²) -3(27))/2 = 1782, so it is the 27th heptagonal number after 0.

Here’s another cool fact about 1782 from OEIS.org.