Question

if m is an even number prove that m^2 is a multiple of 4

Answer 1

Answer:

How do you prove that the product of two even numbers is always a multiple of four?

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So the formal(-ish) definition of “even number” I see most often is something along the lines of “an integer m for which there exists some other integer k such that m = 2*k”. For example, 10 is an even number because there exists an integer that we call 5 such that 10 = 2*5. 11 is not an even number, because to have 11 = 2*k, k has to be a non-integer.

Now, we posit that we have two even numbers. Call them a and b. From our definition above, we know that there are two other integers (let’s just call them c and d for fun) such that a = 2*c and b = 2*d. Then a*b = (2*c)*(2*d). Now we get to the r

Here is a proof.

All numbers are either prime or have prime factors.

A non-prime number can be represented by a list of the prime numbers one can multiply to get that number (e.g. 50 would be 5, 5 and 2). Because of the definition of a prime number, this list cannot be broken down any further, and any such list that did contain composite (non-prime) numbers could, by definition, be broken down into the factors of that number.

An even number is, by definition, a multiple of two.

The lists representing both numbers must each contain a 2. Therefore, the union (combination) of these lists must contain at least two ‘2’s. Thus, the resulting number will be a multiple of 2, and also a multiple of 2 × 2, which is 4. Thus, four must be a factor, or, conversely, the number must be a multiple of 4.

Answer 2

Answer:

1cm on a map represents 4000000

5cm on a map represents =5×4000000=20,000,000‬cm

1km=1000m

1m=100cm

∴1km=1000×100=100000cm

∴100000cm=1km

∴20,000,000‬cm=10000020,000,000‬km=200km

Thus, the actual distance between two towns is 200km.