Divisibility
In This Chapter
- Divisibility rules for numbers under 10
- Using the osculation method to determine divisibility
- The difference between positive and negative osculators
Finding out what numbers divide into another number can be a tedious process. You may spend a lot of time just inputting numbers into the calculator. However, it doesn’t have to be that difficult!
In this chapter, I take you through some common divisibility rules you can apply to your calculations. I also teach you about a method called osculation, which can make determining what a number’s divisible by much simpler.
Divisibility Rules
If you have a divisor that’s less than 10, it’s very easy to find out whether a number can be divided by it.
A number is divisible by 2 if it’s even, meaning the number has to end in 2, 4, 6, 8, or 0.
A number can be divided by 3 if, when you add the digits in the number, that total is divisible by 3. For example, the number 345 is divisible by 3 because its digits add to 12 (3 + 4 + 5), which is a multiple of 3.
A number can be divided by 4 if the last two digits of the number are divisible by 4. For example, in the number 8,732, the last two digits are divisible by 4, meaning the whole number can be divided by 4.
For divisibility by 5, the end digit of the number has to be a 5 or 0. It’s that simple!
Checking for divisibility by 6 is a little more complicated. Because 6 is divisible by 2 and 3, a number must pass the divisibility tests for both of those. For example, the number 12 is divisible by 6 because it’s even (applicable to the rules for 2) and its digits add to a number that can be divided by 3 (applicable to the rules for 3).
A number can be divided by 8 if the last three digits are divisible by 8. For example, in the number 337,312, the 312 is divisible by 8, which means the whole number can be divided by 8.
For 9, if the sum of the digits add up to 9 or a multiple of 9, you can say the number is divisible by it.
And here’s another easy rule—a number is divisible by 10 if it ends in a zero.
But what about 7? That’s where the following method comes in.
The Osculation Method
The osculation method can help you find the divisibility rules for numbers over 10, in particular prime numbers. You apply the osculation method using an osculator. An osculator is a number you can derive from the divisor to find out whether another number is divisible by it. A number has both a positive and negative osculator. Sounds complicated, doesn’t it? Let me carefully walk you through how to find and apply each type.
If your divisor ends in 9, the positive osculator is one more than the digit before the 9. For example, for the number 29, the osculator is 3, because the digit before the 9 is 2, and one more than that is 3. Likewise, in the number 79, the positive osculator is 8, because one more than the previous digit before the 9 is 8.
But how do you find the positive osculator for numbers that don’t end in 9? Depending on the last digit, you multiply so the number ends in 9:
- If a number ends in 1, multiply it by 9.
- If a number ends in 3, multiply it by 3.
- If a number ends in 7, multiply it by 7.
The result of the multiplication gives you a number that ends in 9, which you can use to find the osculator.
For example, if you need to find the osculator for 13, you simply multiply 13 by 3 to get 39. The osculator is one more than the digit before 9; in this case, it’s 4. So the osculator for the number 13 is 4.
QUICK TIP
Positive osculators are typically used for numbers ending in 3 and 9, because these values are smaller than the negative osculators (which I’ll discuss in the next section). For example, the positive osculator for 19 is 2, while the negative osculator is 17. In this case, using the 2 instead of the 17 for calculations is much simpler and can help you avoid errors.
The result of the osculation process, which I walk you through in the examples, should either be the divisor itself or a multiple of the divisor. That’s when you know the dividend is completely divisible by the divisor. If a number starts to repeat, you can say the number is divisible. Another result that indicates the dividend is divisible by the divisor is when the result of the osculation process is zero.
The following examples show you different scenarios of the osculation process using positive osculators.
Example 1
Find out if 112 is divisible by 7.
Step 1: Find the positive osculator for 7. Because the number is 7, multiply by 7: 7 × 7 = 49. The positive osculator is one more than the digit before the 9, so the positive osculator for 7 is 5.
Step 2: Osculate 112 with 5.
11 + (2 × 5) = 21
2 + (1 × 5) = 7
Solution: You end up at 7, so 112 is divisible by 7.
Example 2
Find out if 1,035 is divisible by 23.
Step 1: Find the positive osculator for 23. Because the number ends in 3, multiply by 3: 23 × 3 = 69. The positive osculator is one more than the digit before 9, so the positive osculator for 23 is 7.
Step 2: Osculate 1,035 with 7.
103 + (5 × 7) = 138
13 + (8 × 7) = 69
6 + (9 × 7) = 69
Solution: 69, a multiple of 23, is repeating, so 1,035 is divisible by 23.
Example 3
Find out if 6,308 is divisible by 38.
Step 1: 38 is a composite number made up of 2 and 19, so test for divisibility by both of those numbers. Because 6,308 is even, it’s divisible by 2. To find out if it’s divisible by 19, find the positive osculator for 19, which is 2.
Step 2: Osculate 6,308 with 2.
630 + (8 × 2) = 646
64 + (6 × 2) = 76
7 + (6 × 2) = 19
Solution: 2 and 19, multiples of 38, go into 6,308, so it is divisible by 38.
Example 4
Find out if 334,455 is divisible by 39.
Step 1: Find the positive osculator for 39. Because the number ends in 9, the osculator is only one more than the digit before the 9. So for 39, the osculator is 4.
Step 2: Osculate 334,455 with 4.
33445 + (5 × 4) = 33465
3346 + (5 × 4) = 3366
336 + (6 × 4) = 360
36 + 0 = 36
Solution: 36 is below the divisor, so 334,455 is not divisible by 39.
Negative Osculation
If you have a divisor that ends in 1, the negative osculator is simply the digit before the 1. For example, if the number is 51, the negative osculator is 5—you simply drop the 1. The same is true if the divisor is 81; you just drop the 1 at the end to get the negative osculator: 8.
What if the divisor doesn’t end in 1? Depending on the last digit, you can find the negative osculator by multiplying:
- If a number ends in 3, multiply it by 7.
- If a number ends in 7, multiply it by 3.
- If a number ends in 9, multiply it by 9.
The result of the multiplication gives you a number that ends in 1; you can then drop the 1 to have your negative osculator.
For example, to find the negative osculator of 7, you multiply it by 3, which gives you 21. You then simply drop the 1 to get your negative osculator; for 7, it’s 2.
QUICK TIP
Negative osculators are typically used for numbers ending in 1 and 7, because these values are smaller than their positive osculator versions. For example, for 81, the positive osculator is 73 and the negative osculator is 8. It’s obviously much easier to use 8 than 73 for calculations!
You can say that a dividend is divisible by the divisor even when the result of the osculation process is zero, the divisor itself, or a repetition of a previous result.
The following examples show you different scenarios of the osculation process using negative osculators.
Example 1
Find out if 6,603 divisible by 31.
Step 1: Find the negative osculator for 31. Because the number ends in 1, simply drop the 1 to get the negative osculator, which is 3.
Step 2: Osculate 6,603 with 3.
660 – (3 × 3) = 651
65 – (1 × 3) = 62
6 – (2 × 3) = 0
Solution: You end up at 0, which means 6,603 is divisible by 31.
Example 2
Find out if 11,234 is divisible by 41.
Step 1: Find the negative osculator for 41. Because the number ends in 1, simply drop the 1 to get the negative osculator, which is 4.
Step 2: Osculate 11,234 with 4.
1123 – (4 × 4) = 1107
110 – (7 × 4) = 82
8 – (2 × 4) = 0
Solution: You end up at 0, which indicates 11,234 is divisible by 41.
Example 3
Find out if 2,275 is divisible by 7.
Step 1: Find the negative osculator for 7. Because the number is 7, multiply by 3: 7 × 3 = 21. The negative osculator is the digit before the 1, so for 7, it’s 2.
Step 2: Osculate 2,275 with 2.
227 – (5 × 2) = 217
21 – (7 × 2) = 7
Solution: You end up at 7, which means 2,275 is divisible by 7.
Example 4
Find out if 464,411 is divisible by 71.
Step 1: Find the negative osculator for 71. Because the number ends in 1, drop the 1 to get the osculator. For 71, the negative osculator is 7.
Step 2: Osculate 464,411 by 7.
46441 – (1 × 7) = 46434
4643 – (4 × 7) = 4615
461 – (5 × 7) = 426
42 – (6 × 7) = 0
Solution: You end up at 0, which indicates 464,411 is divisible by 71.
QUICK TIP
The sum of the positive and negative osculators equal the divisor. Therefore, if you already know one of the osculators, you can subtract that value from the divisor to get the other osculator. For example, the negative osculator for 61 is 6. You can then subtract 6 from 61 to get the positive osculator, which is 55.
The Least You Need to Know
- If a divisor is under 10, you can follow its divisibility rules to determine whether it can divide into a number.
- For a divisor that ends in 9, the positive osculator is one more than the digit before the 9.
- For a divisor that ends in 1, the negative osculator is the digit before the 1.