__Divisibility test for 10, 5, 2, 4, 8__

These are the divisibility tests which are done by checking the last \(n\) digits. For \(10,\) \(5,\) or \(2,\) check the last digit. For \(4,\) check the last two digits. For \(8,\) check the last three digits. I recommend teaching the rules for 10, 5, and 2, then 4, and finally, 8. I consider this ordering to be from easiest to hardest.

__Divisibility tests for 3, 9, 11__

These are the divisibility tests which involve adding all the digits, in some way. Start this lesson by teaching the rule for 3. Next, give your students this challenge. Following that, teach the rule for 9. Then give your students this challenge. Lastly, teach the rule for 11. I consider the order 3, 9, 11, to be from easiest to hardest. Here are some resources I've found which may help you teach these rules.

Divisibility by 3:

- The how and why using base ten blocks.
- The how and why for the written method.

Divisibility by 9:

- The how and why using base ten blocks.
- The how and why for the written method.

Divisibility by 11:

- The how and why using base ten blocks.
- The how and why for the written method.

Teaching the base ten block method isn't necessary, but it may help some students.

__Divisibility tests for 6 and 12__

These are the divisibility tests which combine methods previously learned. The test for \(6\) involves checking divisibility by \(2\) and \(3,\) while the test for \(12\) involves checking divisibility by \(4\) and \(3.\) After teaching the divisibility rules for 6 and 12, give your students this challenge.

__Divisibility tests for 7 and 13__

These are the divisibility rules that involve separating the last digit from the other digits, multiplying the last digit, then adding or subtracting the result from the other digits. The rule can be repeated as long as desired. Teach the rules for 7 and 13, which are of this type. In a later course, students will see this type applies for any prime. Later, they will also prove the rules for 7, 13, and possibly some other primes. After teaching the divisibility rules for 7 and 13, give your students the challenges found here and here.

Watch these Khan Academy videos:

Do these Khan Academy exercises:

Next, give your students this challenge:

Poly Plug Pattern by NRICH

Conclude by leading this investigation:

Recamán’s Sequence

by MathPickle

4.OA.B.4: Find all factor pairs for a whole number in the range 1-100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1-100 is a multiple of a given one-digit number. Determine whether a given whole number in the range 1-100 is prime or composite.