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1.8: Number Theory - Mathematics


Many students are often amazed at the many formulas and discoveries in mathematics. They ask "Did someone just sit around and think that up?"

They often think mathematics is just a bunch of facts – that it only has to do with addition, subtraction, multiplication and division of numbers. In other words, they think it's all about arithmetic and being good at doing arithmetic in your head.

On the contrary, mathematics is about ideas. It is especially for those who are interested in asking deep questions, and working to find the answers. It is about thinking creatively, about trying things nobody has done before, about using your imagination.

There is a story about a mathematician, David Hilbert, who noticed that one of his students stopped attending class. When he was told that the student had decided to drop mathematics to become a poet, Hilbert replied, "Good – he did not have enough imagination to become a mathematician."

Digital Roots and Divisibility

It's helpful to understand what is meant by the DIGITAL ROOT of a number because they are used in divisibility tests, and are also used for checking arithmetic problem. A DIGITAL ROOT of a number is one of these digits: 0, 1, 2, 3, 4, 5, 6, 7 or 8.

Definition: The DIGITAL ROOT of a number is the remainder obtained when a number is divided by 9.

Exercise 1

1. Divide each of the following numbers by 9. Then, write the remainder.

a. 25h. 8
b. 48i. 54
c. 53j. 74
d. 829k. 481
e. 5402l. 936
f. 3455m. 8314
g. 47522n. 647

Below is another easier way to find the digital root of a number.

Step 1: Add the individual digits of the number.

Step 2: If there is more than one digit after adding all the digits, repeat this process until you get a single digit. If the final sum is 9, write 0, because 9 and 0 are equivalent in digital roots (since the remainder is a number smaller than 9). That is why the digital root of a number is only one of these digits: 0, 1, 2, 3, 4, 5, 6, 7 or 8. Those are the only possible remainders that can be obtained when a number is divided by 9! It can't be 9. The single digit you finally end up with is the DIGITAL ROOT of the number.

Examples: Find the digital roots of the following numbers.

Example 1

1. 34: Add the digits: 3 + 4 = 7 The digital root of 34 is 7

Example 2

2. 321: Add the digits: 3 + 2 + 1 = 6 The digital root of 321 is 6

Example 3

3. 58: Add the digits: 5 + 8 = 13 Add the digits again: 1 + 3 = 4
The digital root of 58 is 4

Example (PageIndex{1})

4. 97: Add the digits: 9 + 7 = 16 Add the digits again: 1 + 6 = 7
The digital root of 97 is 7

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5. 72: Add the digits: 7 + 2 = 9 In digital roots, 9 is the same as 0
The digital root of 72 is 0

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6. 346,721: Add the digits, we get 3 + 4 + 6 + 7 + 2 + 1= 23
Add the digits again: 2 + 3 = 5 The digital root of 346,725 is 5

Note that you should not write 97 = 16 = 7. 97 IS NOT EQUAL to 7!! The digital root of 97 is equal to 7. I use dashes, colons or arrows to record the digital root, so it looks something like this: 346,721  23  5. Don't use equal signs!!!

Another method for finding digital roots uses the fact that 9 and 0 are equivalent in digital roots. The process of finding digital roots is also called Casting out Nines. When you add the digits, you don't have to add the digit 9 or any combination of numbers that add up to 9 (like 2 and 7, or 5 and 4, or 2 and 3 and 4, etc.) – you can "cast out" all 9's. Cross them off; then add the remaining digits together. The following example illustrates how "casting out nines" simplifies the process of finding the digital root of a large number.

Example: Find the digital root of 5,624,398.

Without casting out nines: 5 + 6 + 2 + 4 + 3 + 9 + 8 = 37. Add again: 3 + 7 = 10. Add again: 1 + 0 = 1.

Casting out Nines: Since 5 and 4 add up to 9, cross them off: . Since 6 and 3 add up to 9, also cross them off: . Also cross off the 9: . The only digits to add are the 2 and 8, which is 10. The digital root of 10 is 1. So 1 is the digital root of 5,624,398 which is the same answer obtained without first casting out nines.

Below, the digital roots for examples 4, 5 and 6 on the previous page is computed again using casting out nines. Note that the digital root remains the same.

Example 4

Example 4. 97: Cross off the 9. Only the 7 remains. The digital root is 7.

Example 5

Example 5. 72: 7 + 2 = 9, so cross them off. Therefore, the digital root is 0.

Example 6

Example 6. 346721: Cross off the 3 and 6, and also the 7 and 2. The only digits to add are the 4 and 1. Therefore, the digital root is 5.

Exercise (PageIndex{1})

2. Find the digital roots of the following numbers, using either method. Remember that 9 is not a digital root. Write zero (0) if the sum is 9 (cast out the 9).

a. 647

Exercise (PageIndex{1})

3. Did you obtain the same answers for exercises 1 and 2?

Note: This method of finding the digital root only has to do with the number 9. You can't find the remainder of a number when dividing by 7 by adding the digits.

Using Digital Roots is to Check Addition and Multiplication Problems.

To check an addition problem, add the digital roots of the addends. Then, check to see if the digital root of that sum is the same as the digital root of the actual sum of the addends. This works whether there are only two addends, or several addends.

Below are some examples of how to check addition. The actual addition is shown to the left. Arrows are used to show the digital roots of the addends and sum. To check, the digital roots of the addends are added together, and then the digital root of that sum is computed. Compare it to the digital root of the actual sum. If they are equal, put a check to indicate the answer is probably correct. NOTE: There is a slight chance that the digital roots match, but the answer is still not correct due to some other mistake, like transposing digits. For instance, in example 1 below, it's possible someone might write down 1153 for the answer. The digital roots in the check would match. The possibility of this happening is slight, so we generally assume the problem was done correctly if the digital roots check. On the other hand, if the digital roots do not check, you know it is wrong.

Example 1

Example 1: 723  3

+ 412  7

1135  1

Ck: 3 + 7 = 10  1 √ Correct!

Explanation: To check, add the digital root of the addends (3 + 7 = 10); then find the digital root of 10 (1). Verify that this equals the digital root of the sum, 1135 (1). Since it does, the addition problem was probably done correctly.

Example 2

Example 2: 463  4

+ 529  7

1630  1

Ck: 4 + 7 = 11  2 Wrong!

Explanation: To check, add the digital root of the addends (4 + 7 = 11); then find the digital root of 11 (2). This should equal the digital root of the sum, 1630 (1). Since it doesn't , there is a mistake and the addition problem was done incorrectly.

Exercise (PageIndex{1})

4. Someone did the following addition problems, but only wrote down the answers. Check the answer to each problem by using digital roots. Note that the procedure is the same if there are more than two addends. Add the digital roots of all the addends. Show Work!

For a multiplication problem, the check is similar, except to check, you multiply the digital roots of the numbers you are multiplying. To check addition, you ADD the digital roots of the addends and check against the digital root of the sum. To check multiplication, you MULTIPLY the digital root of the numbers being multiplied and check against the digital root of the product. Here are some examples that provide an answer someone might have written down after doing the multiplication problem on another piece of paper.

Example 1

Example 1: 85  13 4

x 63  0

5355  18 0

Ck: 4 x 0 = 0 Correct!

Explanation: To check, multiply the digital roots of the numbers you are multiplying (4 x 0 = 0); then find the digital root of 0 (0). Verify that it equals the digital root of the product, 5355 (0). Since it does, the multiplication problem was probably done correctly.

Example 2

Example 2: 43  7

x 52  7

2136  3

Ck: 7 x 7 = 49  4 Incorrect!

Explanation: To check, multiply the digital roots of the numbers you are multiplying (7 x 7 = 49); then find the digital root of 49 (4). This should equal the digital root of the product, 2136 (3). Since it doesn't , there is a mistake. Therefore, the multiplication problem was done incorrectly.

Exercise (PageIndex{1})

5. Someone did the following multiplication problems, but only wrote down the answers. Show Work!

Exercise (PageIndex{1})

6. Add or multiply the following as indicated, then use digital roots to check the answer to each problem. Show Work!

a. 7362

+ 5732

b. 8308 + 956

a. 7362

+ 5732

a. 7362

+ 5732

a. 7362

+ 5732

Using Digital Roots is to Check Subtraction Problems.

To check subtraction, you usually add the difference (the answer) to the subtrahend (the number that was subtracted) and see if you get the minuend (the number you subtracted from). In other words, to check that 215 – 134 = 81 was done correctly, simply add 81 and 134, which is 215. To check subtraction using digital roots, simply add the digital root of the difference to the digital root of the subtrahend and see if you get the digital root of the minuend. To check that 215 – 134 = 81 was done correctly, simply add the digital roots 81 and 134: 0 + 8 = 8. Verify that 8 is the digital root of the minuend, 215. It is! Here is another example. Let's say someone did the following subtraction problem: 5462 – 2873 = 2589. To check, add the digital roots of 2589 and 2873, or 6 + 2 to get 8. Since 8 is also the digital root of 5462, the answer is probably correct.

Here are some more examples of how to check subtraction using digital roots.

Example (PageIndex{1})

Example 1: 7362  0

– 5732  8

1630  1

Ck: 1 + 8 = 9  0 √ Correct!

Explanation: To check, add the digital root of the difference and subtrahend (1 + 8 = 9). The digital root of 9 is 0. Verify that this equals the digital root of the minuend, 7632 (0). Since it does, the subtraction was probably done correctly.

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Example 2: 5073  6

– 878  5

4205  2

Ck: 2 + 5 = 7 WRONG!

Explanation: To check, add the digital root of the difference and subtrahend (2 + 5 = 7). This should equal the digital root of the minuend, 5073 (6). Since it doesn't, the subtraction was not done correctly.

Exercise (PageIndex{1})

7. Someone did the following subtractions problems, but only wrote down the answers. Show Work!

Exercise (PageIndex{1})

8. Subtract, add or multiply the following as indicated, then use digital roots to check the answer to each problem. Show Work!

Using Digital Roots is to Check Division Problems.

To check division, you multiply the divisor (the number you are dividing by) by the quotient (the answer), and then add the remainder. It is correct if the answer obtained is equal to the dividend (the number you divided into). In other words, here is how to check this division, 431 ÷ 34 = 12 r. 23: Multiply 34 by 12 and add 23. Here are the steps: 34 x 12 + 23 = 408 + 23 = 431. Since 431 is the dividend, this problem was done correctly. To check division using digital roots, simply multiply the digital root of the divisor by the digital root of the quotient, and add the digital root of the remainder. If the digital root of that number equals the digital root of the dividend, it is probably correct.

Note: After the digital roots of the divisor and quotient are multiplied, you can first find the digital root of that product before adding the digital root of the remainder.

Here is how to check the division problem we did above by using digital roots. Multiply the digital root of the divisor, 34, by the digital root of the quotient, 12: 7 x 3 = 21. Then add the digital root of the remainder, 23: 21 + 5 = 26. Determine the digital root of that number, 26, which is 8. Check to see if 8 matches the digital root of the dividend, 431. It does. As stated in the note, you could determine the digital root of 21 before adding the digital root of the remainder. In this case, it would have looked like this: 7 x 3 = 213. Then add the digital root of the remainder, 23: 3 + 5 = 8. Verify that this equals the digital root of the dividend, 431. Therefore, the division was probably done correctly. More examples follow. The check without digital roots is shown to the right. The digital root check is shown under the division problem.

Example 1

Example 2

Below is one more example for you to study.

Example 3

Exercise 9

9. Someone did the following division problems, but only wrote down the answers. Show work. You don't need to do the actual check as shown to the right of the last three examples. Show work!

a.

b.

One more note of caution about using digital roots:

In problem 9d, a common mistake some people make is to forget to put the zero in the quotient, 102. Instead, the answer might have been written as 12 r. 32. Using digital roots, the answer would check. So, you should also check the reasonableness of the answer by approximating. For instance, (78 imes 12) is about 80 times 10 or 800, which is not even close to equaling the dividend, 7988. Another time digital roots might fail is when someone transposes the digits of a number. If the answer to a problem was 465, and it was written as 456, using digital roots wouldn't detect the mistake.

Exercise (PageIndex{1})

10. Do the following division problem, and check the answer using digital roots. Show work.

23972 ÷ 156 = ____________

Later in this module, we will explore prime numbers, composite numbers, the greatest common factor of two or more numbers, and the least common multiple of two or more numbers. Factoring is the method used to find the prime factorization of a composite number, and it can also be used to find the greatest common factor and least common multiple of a set of numbers. One of the problems in factoring large numbers is that sometimes it isn't clear if it is prime or composite. In other words, it isn't clear whether it has any factors other than 1 or itself. Most of us know that if the last digit of a numeral is even, then 2 will divide into it; or if it ends in 0, 10 will divide into it; or if it ends in 0 or 5, that 5 will divide into it. Sometimes, people even have trouble determining if relatively small numbers are prime. For instance, many people think 91 is prime, but in fact it is not. Knowing some divisibility tests makes the task easier, so we'll soon take some time to discuss divisibility tests for several numbers. First, we need to go over some notation concerning divisibility.

When you see 12/3, this means 12 "divided by" 3. The slash that slants to the right is another way to write the division sign, ÷. 12/3 (or 12 ÷ 3) is a division problem, and the answer is 4.

Here is something altogether different. If I say "3 divides 12", I am making a statement. "3 divides 12" is not a division problem that needs to be done. It is a statement that happens to be true. The symbol used to represent the word "divides" is a vertical line. So, "3 divides 12" can be written "3|12". Again, this is a statement, a fact, not a division problem. The way to express "does not divide" is to put a slash through the symbol:

So, how do I know "3 divides 12" is a true statement? The definition of "divides" follows:

Definition: "a divides b" if there exists a whole number, n, such that an = b. This is simply saying that "a divides b" means a is a factor of b! In shorthand notation, this is written "a|b if there exists a whole number, n, such that an = b, or a|b means that a is a factor of b."

Okay, so what exactly does "there exists a whole number, n, such that an = b" mean? Well, it means that the first number times some whole number equals the second number. Or you can think "a divides b" is true if (b div a) is a whole number.

Let's get back to why "3 divides 12" is a true statement. Hmmm...you can be formal and ask yourself: Is there a whole number n such that 3n = 12? Or, you can simply ask yourself: Is 3 a factor (or divisor) of 12? In either case, the answer is yes, so the statement is true.

Let's work a little more on the difference between a statement using "divides", and an actual division problem.

Example (PageIndex{1})

Examples: Determine if each of the following is a statement or if it is a division problem. If it is a statement, state if it is true or false and back up your answer. If it is a division problem, state the answer to the division problem.

Example (PageIndex{1})

Example 1: 4|20

Solution: This is a true statement, because 4 is a factor of 20 (since 4 · 5 = 20)

Example 2

Example 2: 20/6

Solution: This is a division problem. The answer is 3 r. 2

Example 3

Example 3: 15|3

Solution: This is a false statement because 15 is not a factor of 3

Example 4

Example 4: 6 divides 20

Solution: This is a false statement because 6 is not a factor of 20

Example 5

Example 5: 3 divides 21

Solution: This is a true statement because 3 is a factor of 21 (since 3 · 7 = 21)

Exercise 11

11. Determine if each of the following is a statement or if it is a division problem. If it is a statement, then decide if it is true or false and back up your answer. If it is a division problem, state the answer to the division problem. Study the examples on the previous page if you need help getting started.

a. 35/7
b. 35|7
c. 7|35:
d. 40/7
e. 56|8
f. 7|40:
g. 12 divides 60
h. 80 divided by 30
i. 70 divided by 5
j. 42 divides 3
k. 6 divides 42
l. 80 divided by 10
m. 100/2


Watch the video: DIVISIBILITY - DISCRETE MATHEMATICS (November 2021).