Determining the divisibility of a number by 37 is a valuable mathematical skill that can simplify calculations and enhance problem-solving abilities. While it may seem like a daunting task, understanding a simple yet effective method can empower you to swiftly ascertain the divisibility of any number by 37. Embark on this mathematical journey and master the art of identifying numbers that are divisible by 37.
To begin, separate the number into two groups of digits: the first group consists of the digits in the odd-numbered positions, starting from the rightmost digit; the second group comprises the digits in the even-numbered positions. Multiply the first group of digits by 3 and the second group of digits by 7. Subtract the second product from the first product. If the result is divisible by 37, then the original number is also divisible by 37.
For instance, consider the number 7,234. The digits in the odd-numbered positions are 7, 3, and 4; their product is 3 x 7 x 3 x 4 = 63. The digits in the even-numbered positions are 2 and 4; their product is 7 x 2 x 4 = 56. Subtracting the second product from the first product, we get 63 – 56 = 7. Since 7 is not divisible by 37, the number 7,234 is not divisible by 37.
The Divisibility Rule for 37
1. Divide the number into pairs of digits, starting from the right:
Divide the number into groups of two digits, starting from the rightmost digit. If there are any digits left over at the end, treat them as a single group.
| Number | Pairs of Digits |
|---|---|
| 4774 | 47 | 74 |
| 31487 | 31 | 48 | 7 |
| 654 | 65 | 4 |
2. Multiply the alternating pairs by 1 and -1:
Starting with the rightmost pair, multiply the pairs of digits by 1 and -1 alternately.
| Number | Pairs of Digits | Multipliers | Modified Pairs |
|---|---|---|---|
| 4774 | 47 | 74 | 1 | -1 | 47 | -74 |
| 31487 | 31 | 48 | 7 | 1 | -1 | -1 | 31 | -48 | -7 |
| 654 | 65 | 4 | 1 | -1 | 65 | -4 |
3. Add up the modified pairs:
Add the modified pairs of digits together.
| Number | Modified Pairs | Sum |
|---|---|---|
| 4774 | 47 | -74 | 47 – 74 = -27 |
| 31487 | 31 | -48 | -7 | 31 – 48 – 7 = -24 |
| 654 | 65 | -4 | 65 – 4 = 61 |
4. Check the divisibility:
If the sum is divisible by 37, then the original number is divisible by 37.
| Number | Modified Pairs | Sum | Divisible by 37 |
|---|---|---|---|
| 4774 | 47 | -74 | -27 | No |
| 31487 | 31 | -48 | -7 | -24 | No |
| 654 | 65 | -4 | 61 | No |
Identifying the Base Case
The first step in determining if a number is divisible by 37 is to identify the base case. This is the number that, when divided by 37, leaves a remainder of 0. For example, 0 is the base case because 0 divided by 37 equals 0.
In general, a number x is divisible by 37 if and only if x - 37 is divisible by 37. This suggests a recursive approach to finding the base case: repeatedly subtract 37 from a number until the result is less than 37. The result of this subtraction is the base case.
For instance, to find the base case of 111, we follow the steps below:
- Subtract 37 from 111:
111 - 37 = 74 - Subtract 37 from 74:
74 - 37 = 37 - Stop because the result is less than 37.
In this example, 37 is the base case of 111.
The Multiplication Step
To perform the multiplication step, we multiply the last two digits of the number by 11. For instance, if the number is 121, the last two digits are 21. We then multiply 21 by 11, resulting in 231.
Number 3
In the case of the number 3, we have the following table:
| Number | Last Two Digits | Multiplication by 11 |
|---|---|---|
| 3 | 3 | 33 |
| 30 | 30 | 330 |
| 300 | 300 | 3300 |
As observed, multiplying the last two digits of any number ending in 3 by 11 always results in a number ending in 3. Consequently, if the result of the multiplication step ends in 3, the original number is divisible by 37.
The Subtraction Step
The subtraction step is a simple but effective way to check if a number is divisible by 37. To perform this step, follow these instructions:
1. Subtract 11 from the number.
For example, if you want to check if 456 is divisible by 37, start by subtracting 11:
“`
456 – 11 = 445
“`
2. If the result is divisible by 37, then the original number is also divisible by 37.
In this case, 445 is divisible by 37 because 445 ÷ 37 = 12. Therefore, we can conclude that 456 is also divisible by 37.
3. If the result is not divisible by 37, continue to Step 4.
4. Subtract 37 from the result of Step 1.
In our example, since 445 is not divisible by 37, we subtract 37:
“`
445 – 37 = 408
“`
5. If the result of Step 4 is divisible by 37, then the original number is also divisible by 37.
In this case, 408 is divisible by 37 because 408 ÷ 37 = 11. Therefore, we can conclude that 456 is also divisible by 37.
6. If the result of Step 4 is not divisible by 37, then the original number is not divisible by 37.
The following table summarizes the steps involved in the subtraction step:
| Step | Action |
|---|---|
| 1 | Subtract 11 |
| 2 | Check if the result is divisible by 37 |
| 3 | If the result is not divisible by 37, subtract 37 |
| 4 | Check if the second result is divisible by 37 |
| 5 | Conclude based on the second result |
Checking the Remainder
Step 1: Find the Remainder
The first step is to find the remainder when you divide the number by 37. To do this, use the modulo operator "%". In Python, the code would look like this:
remainder = number % 37
Step 2: Check the Remainder
Once you have the remainder, check if it is equal to zero. If it is, then the number is divisible by 37. If it is not, then the number is not divisible by 37.
For example:
- If remainder is 0, the number is divisible by 37.
- If remainder is 1-36, the number is not divisible by 37.
Step 3: Handle Special Cases
When you are working with the remainder, there are a few special cases to keep in mind:
- If the remainder is negative, add 37 to get the positive remainder.
- If the remainder is 0, the number is divisible by 37.
- If the remainder is not 0, the number is not divisible by 37.
For example:
| Number | Remainder | Divisible by 37 |
|---|---|---|
| 37 | 0 | Yes |
| 74 | 0 | Yes |
| 111 | 0 | Yes |
| 148 | 34 | No |
| 185 | 19 | No |
| 222 | 2 | No |
Visualizing the Process with an Example
To illustrate the process of determining divisibility by 37, we’ll work through a specific example: the number 153. We start by writing the number as the sum of its individual digits, 1 + 5 + 3.
Step 1: Sum the Digits
The sum of the digits in 153 is 1 + 5 + 3 = 9.
Step 2: Multiply the Sum by 11
We then multiply the sum by 11, which yields 9 × 11 = 99.
Step 3: Subtract the Original Number
Next, we subtract the original number, 153, from the product: 99 – 153 = -54.
Step 4: Take the Absolute Value
Since the result is negative, we take its absolute value: |-54| = 54.
Step 5: Divide by 37
Finally, we divide the absolute value by 37: 54 ÷ 37 = 1 with a remainder of 17.
Step 6: Check the Remainder
Because there is a remainder of 17, we conclude that 153 is not divisible by 37.
You can use this step-by-step method to determine divisibility by 37 for any number you encounter.
Exceptions and Special Cases
When dealing with numbers that may not follow the standard divisibility rules for 37, there are certain exceptions and special cases to consider:
The Number 7
One unique case to note is the number 7. While 37 is not directly divisible by 7, there is a special relationship between them. Multiplying any number ending in 7 by 3 will always result in a number that is divisible by 7.
Example:
3 * 7 = 21 (divisible by 7)
3 * 37 = 111 (divisible by 7)
3 * 47 = 141 (divisible by 7)
This rule also extends to numbers ending in multiples of 7, such as 14, 21, 28, and so on.
Table: Divisibility of Numbers Ending in 7 by 3 and 7
| Number | Multiplied by 3 | Divisible by 7? |
|---|---|---|
| 7 | 3 | Yes |
| 14 | 3 | Yes |
| 21 | 3 | Yes |
| 28 | 3 | Yes |
| 35 | 3 | Yes |
| 42 | 3 | Yes |
| 49 | 3 | Yes |
| … | … | … |
Using the Rule for Larger Numbers
To use this method on larger numbers, you need to break the number down into its component parts. First, identify the last two digits of the number. Then, multiply the first digit of the number by 11. Subtract the result of this multiplication from the last two digits of the number.
If the result of the subtraction is divisible by 37, then the original number is also divisible by 37. If the result is not divisible by 37, then the original number is not divisible by 37.
Example
Let’s try this method on the number 8779. The last two digits of 8779 are 79. The first digit of 8779 is 8. Multiplying 8 by 11 gives us 88. Subtracting 88 from 79 gives us -9. -9 is not divisible by 37, so 8779 is not divisible by 37.
| Number | Last 2 Digits | First Digit | Result of Subtraction | Divisible by 37? |
|---|---|---|---|---|
| 8779 | 79 | 8 | -9 | No |
If the result of the subtraction is 0, then the original number is divisible by 37. If the result of the subtraction is not 0, then the original number is not divisible by 37.
Mathematical Explanation of the Rule
The divisibility rule for 37 involves analyzing the sum of the digits in the number and their alternating products. Let’s denote the number as N.
…
If the absolute difference between the two results is divisible by 37, then N is also divisible by 37. Conversely, if the difference is not divisible by 37, then N is not divisible by 37.
Subtraction of Digits and Products in Action
For a concrete example, let’s take the number 9 = 1089. Breaking it down into its digits:
9081 = 9000 + 80 + 1 = 9081
The individual digits are multiplied as follows:
9 × 1 × 8 × 9 = 648
Then, we apply the rule:
9081 – 648 = 8433
Since 8433 is divisible by 37, we can conclude that 9 is also divisible by 37. To illustrate with an expanded subtraction table:
| Sum of Digits | Product of Digits | Difference | |
|---|---|---|---|
| 1089 | 18 | 72 | -54 |
As the difference (-54) is divisible by 37, we confirm that 9 is divisible by 37.
Divisibility by 10 in Detail
The divisibility rule for 10 states that a number is divisible by 10 if its last digit is 0. This rule is based on the fact that 10 is a factor of 100, which is the smallest power of 10 that contains the last digit of a number.
For example, the number 1234 is divisible by 10 because its last digit is 0. This is because 1234 can be written as 123 * 10 + 0, and 10 is a factor of 123 * 10.
Here is a table that summarizes the divisibility rule for 10:
| Number | Last Digit | Divisible by 10 |
|—|—|—|
| 1234 | 0 | Yes |
| 5678 | 8 | No |
| 9012 | 2 | Yes |
| 34560 | 0 | Yes |
To apply the divisibility rule for 10, simply look at the last digit of the number. If the last digit is 0, then the number is divisible by 10. If the last digit is not 0, then the number is not divisible by 10.
The divisibility rule for 10 is a useful tool for quickly determining whether a number is divisible by 10. This rule can be used to simplify calculations, such as finding the remainder when dividing a number by 10.
How To Tell If A Number Is Divisible By 37
A number is divisible by 37 if and only if the number formed by the last two digits is divisible by 37 or the number formed by the last three digits is divisible by 37.
For example:
- 12345678910 is divisible by 37 because 10 is divisible by 37.
- 12345678911 is divisible by 37 because 11 is divisible by 37.
- 12345678927 is not divisible by 37 because 27 is not divisible by 37.
People Also Ask
Can you calculate how many times 37 goes into a number?
To calculate how many times 37 goes into a number, you can use the following steps:
- Divide the number by 37.
- The quotient will tell you how many times 37 goes into the number.
- The remainder will tell you how many leftover there is.
- Divide the number by 37.
- The remainder will be the number that is left over after you have divided the number by 37.
How do you find the remainder when you divide a number by 37?
To find the remainder when you divide a number by 37, you can use the following steps:
