The Impossible Proof

Today, to have some fun, I'm going to prove something wrong. The challenge today won't be to find the right answer, but to find the flaw in the answer.

Prove: 5 = 7

Well, it looks impossible and wrong at first, but let's try anyway.

Let a = b

Multiply both sides by a and add a² - 2a to both sides:
2a² - 2ab = a² - ab

Take 2 as a common factor:
2(a² - ab) = a² - ab

Divide by a² - ab:
2 = 1

Multiply by 2:
4 = 2

Add 3 to both sides:
7 = 5

By the reflexive property of equality:
5 = 7

Q.E.D.

Obviously, there is something wrong with this proof, and detailed reading will find it. One step was eliminated from this proof. As you can see, after the first assignment of variables (a = b), two steps were done at once. When you break down the steps, you find that a² = ab. This leads to a² - ab = 0, which shows us that the step that led to 2 = 1 was division by zero.

Now you can all understand why the first commandment of math is "Thou shalt not divide by zero". Division by zero can prove anything and destroy mathematic foundations. Only divide by non-zero numbers.

Nadav

nadavs

Multiplication and Remainders

Today's question is a very nice question I found on Yahoo Answers. There was a trigonometry question I was planning for today, but this question is more unique (and besides that, this blog has already many trigonometry questions).

When an integer N is divided by D, it gives a remainder of 7. When N is multiplied by 5 and divided by D, it gives a remainder of 10. What is D?

At first, you may think "I need N to do that". Well, if you had N, the solution would be obvious. That's why we need some creative thinking to find what this number D is.

First, we know that D must be greater than 10. If a/b gives a remainder of m, then b must be greater than m (if you divide something by 7, you can't get a remainder of 8, since that means the result of division must be greater by 1).

Now, since N/D gives a remainder of 7:
N = xD + 7

Also, we know that 5N/D gives a remainder of 10:
5N = yD + 10

Where x and y are integers.

Now multiply the first equation by 5:
5N = 5xD + 35
And equate it to the second one:
5xD + 35 = yD + 10
yD - 5xD = 25
D(y - 5x) = 25
D = 25 / (y - 5x)

Since D must be an integer, 25 must be divided by one of its factors: 1, 5, or 25. However, dividing it by 25 or 5 will give 1 or 5, which is less than 10. For this reason, D must be 25.

You can now take any number for x and create N. For example, let's see what happens when x = 3:
N = 3*25 + 7
N = 75 + 7 = 82
N/25 = 3 with remainder 7 (not surprising, as we went from the definition backwards).

Now multiply this N by 5:
5N = 410
Divide by 25:
410 / 25 = 16 with remainder 10

Nice, isn't it?
Have a great week,
Nadav

nadavs