Thursday, January 3, 2013

Is 2013 a prime year? (A look at divisibility)

Happy New Year!  I hope your year is off to a great start.

If you like to contemplate numbers like I do, you may have wondered whether or not 2013 is a prime number. After a quick review of basic divisibility rules (yeah, there's a song for that) we can see that 2013 is divisible by 3.

(2 + 0 + 1 + 3 = 6, and 3 goes into 6.)

Factoring out 3, we have:  2013 = 3(671)

So, is 671 a prime number?  It doesn’t fit any of the divisibility rules for 2 through 10.  However, as it turns out, 671 is divisible by 11.

Factoring out 11, we have:  671 = 11(61)

Therefore, the prime factorization of 2013 is 3·11·61.

In general, how can we tell whether or not a number is divisible by 11?

Here's a fairly easy Divisibility Rule for 11:

1) Find the sum of every other digit in the number (1st digit + 3rd digit + 5th digit + …)

2) Find the sum of the digits not used in step 1 (2nd digit + 4th digit + 6th digit + …)

If these sums differ by 0, or a multiple of 11, then the original number is divisible by 11.

{If you want to know why this rule works, continue reading.  If you don't care why it works, but would like to see another neat divisibility rule for 11, skip to the bottom.}

Why this rule Works

Think back to elementary school, when you used pencil and paper to multiply.  What does it look like when a number is multiplied by 11?

Here is an example:                 12345
×         11
12345
+  123450
135795

Let’s take a look at a general case, in which a 5-digit number is multiplied by 11.  The conclusions we draw can be generalized for multiplying any length of number by 11.

ABCDE
×           11
ABCDE
+  ABCDE0

The answer will have the following values for its digits:

1st digit:           A
2nd digit:          (A + B)
3rd digit:           (B + C)
4th digit:           (C + D)
5th digit:           (D + E)
6th digit:           E

The sum of digits 1, 3 and 5 is A + (B + C) + (D + E)

The sum of digits 2, 5, and 6 is (A + B) + (C + D) + E

Clearly these sums have a difference of 0.

But… what if we need to carry when adding ABCDE + ABCDE0?

Exactly what happens when we carry?

If a digit’s value exceeds 9, then we subtract 10 from that value, and add 1 to the value of the digit to the left.

For example:               39
+  7
46

(9 + 7 = 16.  16 – 10 = 6.   We write 6 in the one’s place, and add 1 to the digit in the ten’s place)

Subtracting 10 from a digit’s value and adding 1 to the value of the digit to the left causes a net change of 11 between the sum of digits 1, 3 & 5 and the sum of digits 2, 4 & 6.

Let’s assume that in our generalized example, (D + E) exceeds 9.  In this case, since we have to carry, the answer will have the following values for its digits:

1st digit:           A
2nd digit:          (A + B)
3rd digit:           (B + C)
4th digit:           (C + D) + 1
5th digit:           (D + E) – 10
6th digit:           E

Now the sum of digits 1, 3 and 5 is  A + B + C + D + E – 10

And the sum of digits 2, 4 and 6 is  A + B + C + D + E + 1

These sums differ by 11.

Every time we carry (subtract 10 from a digit’s value and add 1 to the value of the digit to the left) we are causing a net change of 11 between the sum of digits 1, 3 & 5 and the sum of digits 2, 4 & 6.  These changes may accumulate, or cancel each other out.

If we carry twice, the sums could differ by 22 (if the net changes of 11 accumulate) or they could differ by 0 (if the net changes cancel each other out).

By generalizing this explanation, we can show that for any multiple of 11, the sum of digits 1, 3, 5, etc. will differ from the sum of digits 2, 4, 6, etc. by 0, or a multiple of 11.

Here’s an even easier divisibility rule for 11:

Subtract the last digit from the number formed by the remaining digits.  (You can repeat this step as often as necessary.)  If the resulting number is divisible by 11, then the original number is divisible by 11.

(Example:  671 is divisible by 11 because 67 – 1 = 66, and 66 is divisible by 11.)

Why does this work?  I bet you can figure it out…

*Hint: It might be helpful to once again consider this problem:

ABCDE
×           11
ABCDE
+  ABCDE0

If you want to try out an explanation, feel free to post a comment below.

Here’s to a great (not to mention, mathtastic) year in 2013.

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