Congratulations — you have reached the final lesson of Divisibility Shortcuts! Over the previous four lessons you built a complete toolkit of quick tests: last-digit checks for $2$, $5$, and $10$; digit-sum checks for $3$ and $9$; a last-two-digits check for $4$; and a combined two-step check for . Each rule let you spot a specific divisor without performing long division. Now it is time to bring and practice running the full set of tests on a single number, so you can quickly identify every divisor from that applies.

Think of all seven rules as tools on a belt. Each one answers a single yes-or-no question about a number, and each one takes only a few seconds. The real power appears when you use them all at once on the same number, because in just a minute or two you can learn a great deal about how that number can be split into equal groups.

Here is the full set for easy reference:

When you need to test a number against all seven divisors, it helps to follow a consistent order rather than jumping around randomly. Here is one efficient sequence:

1. Look at the last digit. This single digit immediately tells you about $2$, $5$, and $10$.
2. Look at the last two digits. This tells you about $4$.
3. Add up all the digits. The digit sum tells you about $3$ and $$.

Let's apply the full scan to the number 2,340.

Step 1 — Last digit. The last digit is $0$. Because $0$ is even, the number is divisible by $2$. Because the last digit is $0$, it also qualifies for $5$ and $10$. Three checks done in a single glance.

As you work through these tests, you will notice that some rules are related but not equivalent. Two pairs are especially important to keep straight.

Divisibility by 2 versus 4. Every number divisible by $4$ is automatically divisible by $2$, because $4 = 2 \times 2$. However, the reverse is not true. Consider $38$: the last digit is $$ (even), so it passes the test for , but remainder , so it fails the test for . Being even means the number has factor of ; divisibility by requires factors of .

Imagine you manage a small shop and a shipment of 504 items arrives. You want to know which group sizes from $\{2, 3, 4, 5, 6, 9, 10\}$ would split the delivery into perfectly equal bundles with nothing left over.

• Last digit: $4$. Even → divisible by $2$. Not or → not divisible by or .

In this lesson you learned how to run all seven divisibility tests on a single number in one efficient pass. By checking the last digit first (for $2$, $5$, $10$), then the last two digits (for $4$), then the digit sum (for $3$ and $9$), and finally combining the and results (for ), you can quickly identify every applicable divisor without performing a single long division. You also saw that passing a simpler rule — like divisibility by — does not automatically guarantee its stricter relative, divisibility by , and the same goes for and .