So, in general, we have the full length, so, note that

this argument applies to any number and in general we have the full length.

Suppose we have some number and we divided by 10 with a remainder,

then the remainder will do just the last digit of our number and

the quotient is the number formed by digit of our initial number,

except the last digit.

So we have the full simple observation and

not that it gives us a very convenient, but also simple corollary.

If you have some integer number e and

it is divisible by 10, if it's last digit is 0, right.

So, note that 10 divides e even only if a remainder is 0 and note that

the remainder is exactly, as we observed, the remainder is exactly the last digit.

So, 10 divides e, even if the last digit is 0.

Okay, now let's consider divisibility by 5.

Okay, let's again start to do the problem, suppose we have

the number 7,347 and the question is, is it divisible by 5?

And we can use the same trick here,

let's break our number into two parts, 7,347 = 734 x 10 + 7.

And now we can break it even more and note that, okay, it

is equal to (734 x 2) x 5 + 5 + 2.

And note that the first two summons are divisible by five and the last one is not.

And note that again, it is the expression for

division of our number by 5 the remainder and here the remainder is 2.

And since the remainder is 2, then our number is not divisible by 5.

Okay and again, we can generalize it.

If you have any integer number it is divisible by 5, if and only if,

it's last digit is 0 or 5.

Okay, so let's give a proof, let's just denote the last digit of a by b.

And let's subtract b from a.

And the resulting number has the last digit 0 and so

it is divisible by 10 and so it is divisible by 5.

So, a minus b is divisible by 5.

And so, as we already shown before, this means that a and

b have the same reminder when we divided them by 5.

So the reminder of a is 0, if and only if,

the reminder of b is 0, so it is divisible by 5, if and

only if, b is divisible by 5, but b is divisible by 5 if b is a digit,

so it's from 0 to 9, it is divisible by 5, if and only if, it is 0 or 5.

So, we have shown our lemma.

Some integer is divisible by 5, if and only if, it's last digit is 0 or 5, so

it's very easy to check whether a given integer is divisible by 5.

Okay and let's also consider divisibility by 2, a very similar idea applies here.

An integer is divisible by 2, if and only if, its last digit is 0, 2, 4, 6, or 8.

In other words, it is divisible by 2, if and

only if, its last digit is divisible by 2.

And again the proof is completely the same, so

let's just denote the last digit of a by b, let's subtract b from a.

And now we have a number, we have a last digit 0, it is clearly divisible by 2.

And so this means that a and

b has the same remainder when we try to divide them by 2.

And so is divisible by 2, if and only if, b is divisible by 2 and

b is divisible by 2 if it is either 0 or 2 or 4 or 6 or 8.

So let's review what we have learned in this lesson.

We are studying number theory and number theory deals with integer numbers and

operations on them.

Number theory is important for fast numerical computations and also,

it is vital for modern cryptography, so this is an important area.

And we have discussed some basic notions divisibility and

remainders and we will use them later on to build more advanced theory.

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