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Now, in this particular session, we would be learning about the multiplicative cipher and the offensive.

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Now the multiplicative cipher is similar to you'll see the cipher, but encrypt using the multiplications

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rather than edition.

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And the afine cipher combines the multiplicative cipher and the Caesar Saiful, resulting in a stronger

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and a more reliable encryption.

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But first, you should learn about the modular arithmetic and the greatest common divisors to mathematical

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concepts that are required to understand and implement the afine cipher.

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So using these concepts, we will create a module to handle a wraparound and find the valid keys for

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the afine safe.

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And we would be using this module when we create a program for a fine cipher in the next session.

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OK, so in this particular session we would be covering modular arithmetic, the modulo operator, the

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greatest common deviser multiple assignment using the aliased algorithm for finding the GCT.

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That is the greatest common deviser, the multiplicative undefined.

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And we will extend the algorithm for finding the modular involves, starting with the modular arithmetic.

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Now the modular arithmetic or the clock arithmetic refers to math in which numbers wrap around when

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they reach a particular value.

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We will use modular arithmetic to handle wraparound in the afine safely.

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Now let's see how we would basically imagine a clock would just on our hand and Otwell replaced with

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a zero.

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If program was designed below, the first one would begin at zero.

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So if the current time is at three o'clock, what time it will be in five?

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That is easy enough to figure out.

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That is three plus five.

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That is eight.

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It will be eight o'clock and five.

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I think of an upper hand starting at three and then moving five hours clockwise.

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If the current time is 10:00, what time it would be in five.

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Also adding five plus 10.

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That is 15.

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But 15 o'clock doesn't make sense for clock.

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That shows only 12.

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So to find out for time, it will be you have to subtract 15 minus 12.

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That is three.

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So it will be three o'clock.

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So normally you would distinguish between three a.m. and three p.m. but that doesn't matter in a modular

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and not double.

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Check this math by moving the other hand clockwise.

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Fiverr will starting something.

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It does indeed land on three.

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If the current time timewasting, what time it will be in two hundred.

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So adding two hundred plus ten is stupid and two then is certainly larger than 12 because one full rotation

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brings the hour hand back to its original position.

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Now we can solve this problem by subtracting 12, which is one rotation until the result is less adding

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to it.

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So subtracting to by 12 is 198 again, 798 still.

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So again, you say continue to subtract 12 until the difference is less, then do it.

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So in this case, the final answer would come to six.

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So if the current time is ten o'clock, the time two hundred hours later would be six o'clock.

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So if you want to double check the ten o'clock plus two hundred hours maddigan repeatedly move the round

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and round the clock face when you move the four hand for both of us, it should have.

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However, it is easier to have the computer do this modular arithmetic for us with the modulo operator.

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So coming to the modulo operator you can use the modulo operator abbreviated as more to write down the

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modular expressions.

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Now in Python, the more operator is the person the site you can think of a more operator as a kind

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of division reminder operator for example, if we see twenty one divided by five comes to four with

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the remainder of one and if we say twenty one modulo five, it comes to one.

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So similarly, if you see 15 modulo 12, it is equal to three.

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Just as 15 o'clock would be three o'clock and falling into the interactive shell to see how the operator

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is coming to us.

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If we go here we type say twenty one, say more below five, you get the value one if we see and plus

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two hundred modulo two if we get sick.

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If we see it then modulated that is zero.

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If we say 20 more below 10 that's also coming to zero, not just as ten o'clock plus two hundred obviously

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wrap 6:00 on the clock with twelve hours.

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That is ten plus two hundred divided by modulo twelve will evaluate.

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Notice that numbers are divided evenly with more to zero suggesting or ten or twenty modulated.

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Later we will use the operator to handle wraparound in the offensive.

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It's also used in the algorithm that we will use to find the greatest common deviser of Poonam now,

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which is enable us to find the valid use for the offensive.

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So moving to finding the fact those two can.

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The greatest common device, no fact, those are the numbers that are multiply to produce a particular

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number considered or foreign to six, that's 20 for now in this equation, four and six are the factors

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of twenty four, because a number of factors can also be used to divide that number without leaving

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any mind.

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So factors are also called devices.

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So number twenty four also has some other factor that is the age into comes through for 12 to two also

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comes to twenty four, twenty four and one also is twenty four.

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So the fact those are for reform would be say one, two, three, four, six, eight, 12 and 20 for

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itself.

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Similarly, if we look at the factors of 30, we may have one in to 32 into 15, three to 10 or say

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five to six.

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The factors of 30 would be say one, two, three, five, 10, 15 and 30.

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So note that any number will have one on itself.

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And it's a factor because one time the number is equal to that number.

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Notice, too, that the list of factors of twenty four and thirty one, two, three, six income.

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So the greatest of these common factors is six or six is the greatest common factor, more commonly

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known as the D.C. the of twenty four.

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And now it is easier to find the good of two numbers by visualizing the effect we will visualize fractals

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and the GCD is using some some manual approach also.

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OK.