while hovering angels touch the string,

Burrowing Owl (Athene cunicularia), family Strigidae, order Strigiformes, Southern CA, USA

It was dreamlike. Nightmarish? Not always. Sometimes it was beautiful.

But unlike the adult in this strip, Calvin’s dad actually APOLOGIZED for yelling at him when he saw how guilty his kid was:

Convoy near Kita, Mali

DIRT documents: [# 5]

Drift stitching is a repair method where zip ties, wires, or clips are used to hold cracked parts together after an accident.

The technique became popular in drifting and motorsports because damaged bumpers could be repaired quickly without replacing the entire panel. Small holes are drilled along the crack so the material can be stitched back together and reinforced.

Over time, the repair style spread beyond racing and became common in DIY car culture. People have also used this method on motorcycles and anything in need of a temporary fix that doesn’t immediately hurt the wallet.

Yes this is one of my fav groups of animals!

Herps and Birds (and More) — Eiselt's Caecilian aka "Penis Snake" Caecilian...

Ringed Caecilian (Siphonops annulatus), family Siphonopidae, widely distributed in South America, east of the Andes

Caecilians are amphibians, like frogs and salamanders. (They are not worms, nor are the snakes).

Most species are fossorial (burrowing), but many are also aquatic or semi-aquatic.

photograph by Carlos Jared

Kaup’s Caecilian (Potomotyphlus kaupii), family Typhlonectidae, found in northern South America

This species is aquatic.

photograph by Dante Fenolio

Beddome’s Caecilian (Ichthyophis beddomei), family Ichthyophiidae, western India

photograph by Girish Gowda

Magdalena Valley Caecilian (Caecilia subnigricans), family Caeciliiidae, San Jacinto, Bolivar, Colombia

photograph by José Gabriel Julio Guzmán 

São Tomé Caecilian (Schistometopum thomense), family Dermophiidae, endemic to São Tomé and Ilhéu das Rolas (Africa)

This is a given.

If the last digit is divisible by 2 (a.k.a. even), then so is the whole number.

If the sum of the digits is divisible by 3, then so is the whole number. The recursivity of this means that if the sum has multiple digits, you can add them up again until you get a single digit and see if it's 3, 6, or 9.

Like the rule for 2, but check if the last two digits are divisible by 4.

If it ends in 5 or 0.

If the rules for both 2 and 3 apply.

No shortcut. Alas.

Like the rules for 2 and 4, but check if the last three digits are divisible by 8. (Yes, this pattern keeps going for 16, 32, etc.)

Like the rule for 3, but the sum of the digits (or the sum of the sum of the digits, etc.) must be 9.

OK, here we go!

Assume our number is a multiple of 7 - i.e. it's congruent to 0 mod 7. I'm using letters to represent the digits here, but there could be an arbitrarily large number of digits, hence the ellipsis.

Each digit is multiplied by a progressively higher power of 10.

Now 10 is congruent to 3 mod 7, so we can replace all the 10's with 3's.

A bit of rearranging gives us this way of expressing the last digit a in terms of the other digits.

When we take all the digits except the last one and move them down one place, we're dividing everything by 3.

Here's where 5 being the multiplicative inverse of 3 mod 7 comes in. Dividing by 3 is the same as multiplying by 5.

Speaking of multiplying by 5, let's do that to our alternative expression for a, and add it on.

Everything cancels out and we get 0.

So if our original number was a multiple of 7, doing this trick will also give us a multiple of 7.

"What about the converse?" you might ask.

If our original number isn't divisible by 7, we get a copy of the remainder in our expression for a. Then instead of cancelling down to 0, we'll be left with 5 times the remainder. This won't be 0 because 7 is prime, and so no non-zero numbers will multiply to make 0 mod 7.