The last digit decides 2
A number divides by 2 when its last digit is even: 0, 2, 4, 6 or 8. 3,746 ends in 6, so it divides by 2. 3,747 ends in 7, so it does not.
The reason is place value. 3,746 is 374 tens and 6 ones. Every ten divides by 2, because 10 = 2 × 5, so all 374 tens together divide by 2, whatever their number. Only the ones digit can leave something over, and 6 ones share into 2 exactly.
The numbers that divide by 2 are ringed. Every one of them ends in 0, 2, 4, 6 or 8.
5 and 10
10 divides by 5 and by 10 as well, so the tens of any number divide by 5 and by 10, and again only the last digit decides.
A number divides by 5 when it ends in 0 or 5, and by 10 only when it ends in 0. 85 ends in 5, so it divides by 5, but it does not divide by 10. 90 ends in 0, so it divides by both.
The numbers that divide by 5 are ringed, and they end in 0 or 5. Only 10, 20 and 30, which end in 0, divide by 10.
Add the digits for 3
The last digit cannot decide 3. 13 ends in 3, but 3 does not divide 13, and 27 ends in 7, but 27 = 3 × 9.
Instead, add all the digits. If that total divides by 3, so does the number. For 471, add 4 + 7 + 1 = 12. 12 divides by 3, so 471 divides by 3, and dividing checks it: 471 = 3 × 157. For 472 the digits add to 13, which does not divide by 3, so 472 does not either.
Why adding the digits works
Each hundred is 99 + 1, and each ten is 9 + 1. So 471 is 4 lots of 99, 7 lots of 9, and then 4 + 7 + 1 ones left over.
99 and 9 both divide by 3, so all the 99s and 9s together divide by 3. What is left over is 4 + 7 + 1, the digit sum. So 471 divides by 3 exactly when its digit sum does.
99 and 9 also divide by 9, so the same test works for 9: a number divides by 9 when its digit sum does. 12 does not divide by 9, so 471 does not.
471 = 100a + 10b + c: every hundred is 99 + 1 and every ten is 9 + 1, so unbundle them and see what 9 cannot swallow
Make 342 and unbundle it
Set the hundreds, tens and ones, then unbundle. Each hundred splits into 99 and 1, and each ten into 9 and 1. The 99s and 9s go into the box, and every number in it divides by 9 and by 3. The ones left outside are the digit sum.
Three slips
Using the last digit to test for 3. The last digit decides only 2, 5 and 10.
Mixing up the tests for 5 and 10. Ending in 0 or 5 is the test for 5. 15 ends in 5, and 10 does not divide 15.
Using odd and even for every test. Being even only shows that a number divides by 2. 14 is even, and neither 5 nor 10 divides it.