Hi there. Let's look at how to add up numbers in modular arithmetic. The key thing to bear in mind is the meaning behind all those symbols we use in modular arithmetic. We are always using the remainders of division and when we adding numbers mod, say mod five, we are adding the remainders of division by five. So, just going to show you a different example. An example that's apparently doesn't have anything to do with modular arithmetic. So, bear with me in this. Say I'm packing oranges in trays, trays that take 12 oranges each. Okay. So, I got my happy oranges here in a tray of 12. So, the 12 oranges really good. I got one tray with 12 oranges. I've given the job that this week I'm going to be packing 140 oranges in trays of 12, okay? So, I've got 140 oranges to pack in trays of 12. How many trays do I get and how many oranges are left unpacked? Well, 140 oranges fills up 11 trays of 12. So, 140 can make 11 full trays of 12 and that brings me to a 132 oranges. Then I've got eight left that don't fill up the full tray. That eight is the remainder of division of 140 by 12. Hundred and forty is eight mod 12. Okay. Say the now on the second week, I've done such a great job packing oranges that I've got 150 to pack same size of trays. So, 150 that's going to be, I can make 12 trays of 12 and that gives me 144 oranges. Then I've got six left that don't make a full tray. That's because 150 is convert with six mod 12. Now, the question is, over the two weeks, how many oranges are not in a full tray? So, you pack them all in trays 12, how many are left in a half filled tray? So, it all boils down to 140 plus 150, what's the remainder of division by 12? We could add 140 plus 150 and then divide by 12 and find a remainder, but we have the remainders of 140 and 150 already calculated, because we had 140 had remainder. Let's have a look, it was remainder eight mod 12 and 150 has remainder, it was remainder six mod 12. Therefore, when you add that up, the 140 add 150 is going to be congruent with eight add six. Basically, you're going to try to pack the eight oranges left and the six oranges left into another tray of 12 and that gives you 14 oranges. Make one tray of 12 plus two leftovers, which means remainder two mod 12, okay? So, we don't really need to add up 140 with 150 and then do the division if we had the partials already done. So, the answer over there is that you get remainder two mod 12. Now, that example just to illustrate the general rule that it's true in modular arithmetic. If you add up, if you have a number, let's call it A and we know that has remainder c when dividing by a number n and you have another number b that has remainder d, when divided by the same n, it is true that the addition of the two numbers a and b has the same remainder as the adding up the two remainders just like we did in the previous page with a being 140 and b being 150. Now, of course, this number here may not be a small enough number, but you can then find the remainder of this new number that it's smaller mod n. Let's have a look at how we use this, just one another example. We have 1,456 and we want to find the remainder of this mod 12. So, how many, what's left after packing your oranges in trays of 12? If you spot that this number can be written as a combination of multiple of 12 plus something else, you can use that to shortcuts the calculations. So, I spot that this number is close to 1,440 which I know is 120 times 12. Because 12 times 12 is 144 and then I've got 16 there. I can use this breakdown to speed up the calculation. A thousand four hundred and forty being a multiple of 12 is congruent with zero mod 12. I'm now down to adding up these and 16 is congruent with four mod 12. Okay. So, you cannot cough multiples of the mod and take them way to speed up your modular arithmetic calculations. One more example. If you are explicitly asked to calculate 256 add 18, still mod 12. We can do the partials again. 256 that is congruent with four mod 12. You can check that out. Then 18 is 12 plus six, so six mod 12. The addition of these guys is going to be congruent with 10 mod 12. Okay. There you are. We write there congruent with 10 mod 12.