Thursday, October 07, 2021

What's so special about 6987952709035199279905674?

√Čric Angelini came up with this yesterday:

abcdefghij... (letters are distinct digits):
a is divisible by 1 with remainder 0
ab is divisible by 2 with remainder 1
abc is divisible by 3 with remainder 2
abcd is divisible by 4 with remainder 3
etc.

The largest such integer would be 95674103:

9/1 leaves a remainder of 0
95/2 leaves a remainder of 1
956/3 leaves a remainder of 2
9567/4 leaves a remainder of 3
95674/5 leaves a remainder of 4
956741/6 leaves a remainder of 5
9567410/7 leaves a remainder of 6
95674103/8 leaves a remainder of 7

Much more interesting is if we dispense with the "distinct digits" limitation. Let us allow any digit. The largest integer is now 6987952709035199279905674:

6/1 leaves a remainder of 0
69/2 leaves a remainder of 1
698/3 leaves a remainder of 2
6987/4 leaves a remainder of 3
69879/5 leaves a remainder of 4
698795/6 leaves a remainder of 5
6987952/7 leaves a remainder of 6
69879527/8 leaves a remainder of 7
698795270/9 leaves a remainder of 8
6987952709/10 leaves a remainder of 9
69879527090/11 leaves a remainder of 10
698795270903/12 leaves a remainder of 11
6987952709035/13 leaves a remainder of 12
69879527090351/14 leaves a remainder of 13
698795270903519/15 leaves a remainder of 14
6987952709035199/16 leaves a remainder of 15
69879527090351992/17 leaves a remainder of 16
698795270903519927/18 leaves a remainder of 17
6987952709035199279/19 leaves a remainder of 18
69879527090351992799/20 leaves a remainder of 19
698795270903519927990/21 leaves a remainder of 20
6987952709035199279905/22 leaves a remainder of 21
69879527090351992799056/23 leaves a remainder of 22
698795270903519927990567/24 leaves a remainder of 23
6987952709035199279905674/25 leaves a remainder of 24

Wednesday, October 06, 2021

Oaklands

click to enlarge
This is my first-year high school class of 1964/1965. Last Saturday, one of these teenagers hosted a high school reunion celebration in his residence at 50 Yorkville Avenue. If one were to look roughly northwest from the top of that building, they might see something like this:

click to enlarge
I have marked the De La Salle college location in yellow. This Apple maps representation is ~2018 because construction on the townhouses (just west of Del's fussball pitch) on Avenue Road is in its starting phase. Additionally, there are now townhouses to the south of the field on Oaklands Avenue. The city of Toronto did a detailed heritage property background report in 2015. Here's a closer bird's-eye view:

click to enlarge
There is a worthwhile article on De La Salle, written in 1980 by William Stephenson, called All Things Right & Dutiful. Somewhat surprisingly for such fare, it contains a paragraph on the once-upon-a-time corporal punishment meted out to some of the students.

I was unfortunately unable to attend this 52nd anniversary (counting from our graduating year) get-together. Many of my reasons noted here were of course tongue-in-cheek.

Saturday, October 02, 2021

Top ten reasons why I am not attending my high-school reunion

#10. I don't want to belong to any club that will accept me as a member. [Groucho Marx]

#9. I'm still recovering from the last time I was talked into attending a get-together.

#8. I can't think of a bigger social evaluative threat. (Not that I care.)

#7. I am under house arrest!

#6. I flipped a coin and it came up tails.

#5. The delta-driven fourth wave (and other respiratory delights).

#4. Elbow-bump performance anxiety.

#3. My dog wasn't invited.

#2. My shoes are on their last legs.

#1. Mama told me not to come. (That ain't the way to have fun.)