 Hello and welcome to the session. In this session first we discuss numbers in general form. Let's try and explore the numbers in detail. Now if you have a two-digit number say AB which is made up of digits A and B, this can be written as 10 into A plus B that is 10A plus B and a two-digit number BA would be written as 10 into B plus A that is 10B plus A. So this is how we can write a two-digit number in a general form. We should remember one thing that here AB does not mean that it is A multiplied by B. In the same way let's consider a three-digit number ABC it is made up of the digits AB and C. This is written as 100 into A plus 10 into B plus 1 into C that is we have the number ABC is equal to 100A plus 10B plus C. In the same way we can change the order of the digits in this three-digit number and write it in the general form. Like if you consider the number 21 let's write this in the general form. This could be written as 10 into 2 plus 1 into 1. Let's consider a three-digit number 1, 2, 1, 121. Let's write this in general form. This would be equal to 100 into 1 plus 10 into 2 plus 1 into 1. Next we discuss games with numbers. First we have reversing the digits two-digit number. Let's see what would be the first trick in this case. In this first the person chooses the number that is a two-digit number AB. Now this can be written as 10A plus B then on reversing the digits we get BA this is written as 10B plus A. Then in the next step we add both these numbers that is we have AB plus BA equal to 10A plus B plus 10B plus A and this is equal to 11A plus 11B that is we get 11 into A plus B. Thus we get the sum of the two numbers the original number that is AB and the number obtained by reversing the digits that is BA is always a multiple of 11. These are divisible by 9. Now when we divide the sum of the two numbers by 11 we get the question as A plus B which is the sum of the two digits of the number AB. Let's try and check this trick using the two-digit number 21. Now this 21 is written as 10 into 2 plus 1 that is 20 plus 1. Now on reversing the digits we get the number as 12 and this could be written as 10 into 1 plus 2 that is 10 plus 2. Now we add both these numbers that is 21 plus 12 would be equal to 20 plus 1 plus 10 plus 2 and this would be equal to 30 plus 3 or you can say 33. Now this 33 as you know that is a multiple of 11 so on dividing 33 by 11 we get 3 and this 3 is the sum of the two digits of the given number 21 that is it is the sum of 2 and 1. Now let's check out the second trick. Let's see how we do this. Now again the person chooses a two-digit number say AB equal to 10A plus B then on reversing the digits the number that we get would be BA which is equal to 10B plus A then in the next step we need to subtract the smaller number out of these two numbers from the larger number. We have to keep some points in mind before doing so let's see. Now if the 10th digit of the given number AB is larger than the 1st digit that is we have A is greater than B then we have 10A plus B minus 10B plus A that is the number BA that is 10B plus A is smaller than the number 10A plus B that is AB and this would be equal to 10A plus B minus 10B minus A and this is equal to 9A minus 9B that is we get 9 into A minus B. If we have that the 1st digit that is B is larger than the 10th digit that is A then 10B plus A minus 10A plus B that is the number AB would be smaller than the number BA and this would be equal to 9 into B minus A. If we have that the 1st digit and the 10th digit are the same then we get 0 as the result after subtracting the numbers. Now as you can see in all the three cases the result that we have obtained that is 9 into A minus B 9 into B minus A and 0 that's when we divide the resulting number that is 9 into A minus B by 9 we get A minus B according as we have A is greater than B and when we divide 9 into B minus A by 9 we get B minus A accordingly when we have A is less than B. Let's try out this trick using this number 21 now this 21 is written as 10 into 2 plus 1 that is 20 plus 1 now on reversing the digits we get the number 12 and this is equal to 10 into 1 plus 2 that is 10 plus 2 now we need to subtract the smaller number from the larger number for that we need to consider the 10th digit and the 1st digit of the given number. Now as you can see here we have the 10th digit is more than or you can say is larger than the 1st digit so we have this number would be smaller than this number so we subtract the smaller number from the larger number so we get 21 minus 12 is equal to 20 plus 1 minus 10 plus 2 and this would be equal to 10 minus 1 and that is equal to 9. Now this 9 as you know is obviously divisible by 9 when we divide 9 by 9 we get 1 this would be equal to 2 minus 1 that is 2 is the 10th digit of the given number and 1 is the 1st digit of the given number and since we have the 2 is greater than 1 so we do 2 minus 1. Next we have reversing the digits 3 digit number let's see how this trick works now first we choose a 3 digit number say a b c this is equal to 100 a plus 10 b plus c now after reversing the order of the digits we get the number c b a this is equal to 100 c plus 10 b plus a now we need to subtract these numbers so we have to keep some points in mind if we have that a is greater than c then the difference between the numbers is given by 100 a plus 10 b plus c minus 100 c plus 10 b plus a and this would be equal to 99 a minus 99 c which is equal to 99 into a minus c then if we have that c is greater than a then the difference between the numbers is given by 100 c plus 10 b plus a minus 100 a plus 10 b plus c and this would be equal to 99 c minus 99 a and that is equal to 99 into c minus a if in case we get that a would be equal to c then the difference is 0 now in each case as you can see the resulting number is divisible by 99 that is when we divide 99 into a minus c by 99 we get the equation as a minus c and 99 into c minus a when divided by 99 gives us c minus a consider a three digit number 125 this is equal to 100 into 1 plus 10 into 2 plus 5 that is 100 plus 20 plus 5 now on reversing the order of the digits we get the number as 521 this is equal to 100 into 5 plus 10 into 2 plus 1 that is 500 plus 20 plus 1 now as you can see here 5 is greater than 1 that is we have c is greater than a then the difference would be given by 521 minus 125 and this is equal to 396 this number 396 is obviously divisible by 99 when we divide 396 by 99 we get 4 which is equal to 5 minus 1 that is difference of the digits 5 and 1 next we discuss forming three digit numbers with given three digits consider the three digits a b and c now we have the number a b c using the three digits a b and c now we can form two more three digit numbers using these three digits a b and c one number that we can form would be c a b now this is formed by shifting the ones digit that is this c to the left end of the number and the other number that we formed would be b c a this number is formed by shifting the hundreds digit that is this a in this case to the right end of the number let's write these three numbers a b c c a b and b c a in general form so we have a b c is equal to 100 a plus 10 b plus c c a b is equal to 100 c plus 10 a plus b then b c a is equal to 100 b plus 10 c plus a now when we add these three numbers that is we have a b c plus c a b plus b c a now this would give us 111 into a plus b plus c that is this could be written as 37 into 3 into a plus b plus c now this is divisible by 37 so when we divide this number that is some of the three numbers by 37 we get the remainder as 0 let's consider the number 125 the digits in this number are 1 2 and 5 now we make two more three digit numbers using these three digits now one number that we obtain would be 512 and the other number that we obtain is 251 so let's add these three numbers that is 125 plus 512 plus 251 this gives us 888 now 888 is divisible by 37 when we divide 888 by 37 we get 24 as the question as we remainder would be 0 next we have letters for digits here we shall deal with the puzzles in which the letters take the place of the digits in an arithmetic sum and we find out which letter represents which digits we will only deal with the problem source addition and multiplication we have few rules to follow while solving such problems or such puzzles the first one says that each letter in the puzzle must stand for just one digit each digit must be represented by just one letter then the other rule is the first digit of a number cannot be 0 and also one more thing that the puzzle must have just one answer consider this in this we have two letters a and b we need to find the values of the letters a and b in this let's consider the addition in the ones column from here we have a plus five gives us two that is a number whose one digit is two and this can happen only when we have a equal to seven since we have seven plus five is equal to 12 and in this as you can see one stitch it is two thus from here we get three seven two five two and on doing this we get b equal to six thus we get a equal to seven and b equal to six this is how we can find the values of the letters in the problems of addition and multiplication this completes the session hope you have understood how we write the numbers in general form then gains with numbers and the letters for digits