Divisibility tests

One night while I was home over the holidays, I ended up helping Emma with her homework. First some spelling homework and then maths. She was confused about some problems on the distributive property and she also had to draw three rectangles with area 24, without using 1 x 24. She didn't have any sort of systematic method; she just cast about for integers that divide 24. She had come up with 6 x 4 and 3 x 8 but couldn't find a third rectangle. So I taught her about prime numbers and prime factorisation.

Somehow this led us to divisibility tests. She knew the one for 2 (and I assume she knows the similar tests for 5 and 10). Most people know these tests. My dad was making nachos (for some reason we were doing her homework on the kitchen floor) and overheard me teaching her the divisibility test for 4. He got all indignant: "I didn't know that! Why don't they teach us cool things like that in school?" and ran into the other room, from where we heard his shouts to Kathi, "Did you know that?!?"

The divisibility test for 4 has the same justification as the test for 2 (or, for that matter, for 5 or 10). For 2, you can write your (positive, for the sake of argument) lengthy integer as 10X + Y where Y is less than 10 (and greater than or equal to 0). Then, if Y is divisible by 2 (so Y = 2Y') we have

10X + Y = 2(5X) + 2Y' = 2(5X + Y')
so the big integer is also divisible by 2. (Note that this works because of the distributive property mentioned earlier.) And, likewise, if Y is not even then the big long integer is also not even. For 4, you instead write your integer as 100X + Y, where Y is now less than 100. Since 4 divides 100, you only have to check whether 4 divides Y (so now you have to check whether 4 divides a 2 digit number, which is more work than the even/odd test, but considerably less work than trying to divide 4 into, say, a 7 digit long number).

So Emma tested some big long numbers for divisibility by 4 and after we found a few that were not divisible by 4, I had her come up with a big long number that was divisible by 4. She ended her number with 08, which I liked.

Then I taught her the divisibility test for 3. If you add up the digits of the number and that sum is divisible by 3, then the whole number is divisible by 3. Rather than give a proof, which is much cleaner and nicer if you know about modular arithmetic, I will just give an example. This also uses the distributive property Emma was working on.

635,148 = 6(100,000) + 3(10,000) + 5(1,000) + 1(100) + 4(10) + 8

=6(1+99,999) + 3(1+9,999) + 5(1+999) + 1(1+99) + 4(1+9) + 8

=6+3+5+1+4+8 + 6(99,999) + 3(9,999) + 5(999) + 1(99) + 4(9)

=(6+3+5+1+4+8) + 3[6(33,333) + 3(3,333) + 5(333) + 1(33) + 4(3)]

So, since the right hand part of the last line (the 3[6(33,333) + 3(3,333) + 5(333) + 1(33) + 4(3)] part) is divisible by 3, that means the whole number is divisible by 3 if and only if the left hand part of the last line (the (6+3+5+1+4+8) part) is also divisible by 3. Since this left hand part is simply the sum of the digits of the number, we get our divisibility test. Of course the same argument works to show that the divisibility by 9 test is analogous: an integer is divisible by 9 if and only if the sum of the digits is divisible by 9.

Calendar

Jan 20: Bach recital at Royal Festival Hall

Feb 22: Mozart's "The Magic Flute" at Royal Opera House

Mar 3:   Strauss' "Salome" at Royal Opera House

Apr 2:    "Dirty Dancing" at Aldwych Theatre

Apr 14:  Bizet's "Carmen" at Royal Opera House

"Gurus, who, I trust you know, are Indian teachers, expect you to contemplate the acorn ten years at a stretch, and if, in that time, you are no wiser about the nut, you are not very bright, and that may be the only certainty with which you will come away, which is a post-graduate melancholy—for no man can find a greater truth than his kidney will allow."
-Djuna Barnes, Nightwood

Granny Squares

When I skyped the Thanksgiving family gathering from London in November, my dad bravely passed his laptop around so that various members of the family could have the disconcerting experience of speaking on a headset to a laptop in a room jam-packed with boisterous relatives. When I talked to Chrissy, she informed me that she had started knitting granny squares to make a blanket for me. Having brought the project with her to Thanksgiving, she was besieged by offers of help, including many from people who had no idea how to knit.

After Thanksgiving, more stories trickled my way. Everyone was helping. Any time someone went into town, cries went up to buy another pair of knitting needles. Those who at first refused to learn to knit (Petie, Matt) resigned themselves to fate by the end of the weekend. The experts (Chrissy, Monica, Marty, Grandma Noble, Kathy (Ray) Torrey, others?) taught the beginners. Kate taught Joanna one morning, only to find Joanna teaching someone else later in the day. Kathy (Ray) Torrey knitted squares even though she's allergic to the knitting needles. When the hardcore folks remaining for that last Saturday night were later asked what they had done, they reported that they had all spent the evening knitting. To the incredulous "wasn't there even any music?" the reply was generally, "no, not that I remember." That last night, Chrissy had laid out the finished squares, placing yarn and knitting needles in the remaining empty blocks so that people could finish squares at home and mail them to Chrissy.

While in New York just before Christmas, Kate got a call from Chrissy on her way up from Baltimore. We agreed to stop by their house to visit the following day, before we went to Connecticut. Later, we found out that we had to be in Connecticut much earlier than we had realized, so we decided to visit the Robin Torreys in Brooklyn that night instead. When we got there, Kate demanded "Do you have It?!?" of Chrissy, who cried back, "I was supposed to have twelve more hours!" (People didn't know whether I knew about the blanket, and didn't want to risk spoiling the surprise.) So that night I saw only the box in which it then sat, but didn't see the squares, which were all finished, but not yet put together.

For the rest of my time at home, everyone I saw who found out that I had seen Chrissy in New York asked eagerly, "Did she give you Something?" I spent the rest of my time at home in Connecticut, the Catskills and Ithaca. Unfortunately, I didn't get to see the Robin Torreys again, and Chrissy had to go back down to Baltimore. The night before I left to return to London, I went back to New York, to Barbara's house. Dad and Kathi met me there but first, unbeknownst to me, stopped at the Robin Torreys' to pick up the blanket.


So here we are at Barbara's house with the beautiful blanket stretched out (those are Linda's feet at the bottom). Some more photos are here.

Thank you all so much! I love it. If you have more photos or stories, I'd love to have them.

The blanket arrived with a warning from Chrissy that it still needed to be "tucked." I had no idea what this meant, so Grandma made Barbara pull out Briana's knitting needles (I believe Briana was one of the many taught at Thanksgiving) so she could teach me to tuck. Grandma then spent half an hour lovingly tucking for me.