# mathematics #mathpuzzle #maths ABCD+ABCD+ABCD+ABCD= DCBA

Hello friends in this video we are going to discuss a beautiful mathematical problem which is find the four digit number a b c d such that four times a b c d will be reverse the a b c d which is d c d a to solve this problem we will use some observation the first observation that a b c d and b c b a are four digit numbers therefore a and d cannot be zero next part d plus d plus d plus d is equal to four d four d is a multiple of four and a is the unit place of four d therefore the possibility of unit place a when the number is multiplied by 4 are 1 into 4 will be 4 the possibility of a may be 4 2 into 4 is 8 the possibility of a is 8 3 into 4 is 12 and a is at the unit place the possibility of a is 2 4 into 4 is 16 the possibility of a is 16 5 into 4 is 20 so possibility of a is 0 but we already observe that a cannot be 0 6 into 4 is 24 so possibility of a is 4 7 into 4 is 28 possibility of a is 8 8 into 4 is 32 so possibility of a is 32 9 into 4 is 36 so possibility of a is if we summarize then a has 4 possibility 2 4 6 8 since d c b a is the four digit number therefore d is the single digit number suppose d is not single digit number suppose d is equal to 12 then d c b a will be the five digit number which is not possible therefore d should be single digit number suppose k is the forwarded carry from the previous sum b plus b plus b plus then we have equation a plus a plus a plus k equal to d now if we check some possibility of a suppose a is equal to 1 then we have 1 plus 1 plus 1 plus 1 plus k equal to d that means 4 plus k equal to d second possibility 2 plus 2 plus 2 plus 2 plus k equal to d that means 8 plus k equal to d next one 3 plus 3 plus 3 plus 3 plus k equal to the that means 12 plus k equal to d which is not possible because t will be two digit number and we already observe that d should be single digit number if we take a equal to 4 then we have 16 plus k equal to d that means d will be the two digit number so it is also not possible and with the similar argument 5 6 7 8 and 9 are not possible therefore a has only two possibility 1 and 2 but we already discussed that a has only four possibility 2 4 6 8 so it cannot be 1 hence a is equal to and so we get the first number a equal to 2 now substitute a equal to in the equation then we have 8 plus k equal to d is a single digit number so k has only two possibility 0 and 1 if k equal to 0 then d equal to 8 and if k equal to 1 d is equal to 9 suppose k is equal to 1 and d is equal to 9 therefore d plus d plus d plus d is equal to 36 where unit digit is 6 but in the sum of d the unit digits should be 2 so d plus d plus d plus d equal to 36 is not possible that means d equal to 9 is not possible and k equal to 1 is not possible and hence d should be 8 and k is equal to 0 that means there is no forwarded carry on a now d plus d plus d plus d is equal to 8 plus 8 plus 8 equal to 32 and 3 is the forwarded carry on c therefore we have another equation 3 plus c plus c plus c plus c where sum has unit digit b therefore we have tan m plus b and hence we have 3 plus 4 c is equal to tan m plus b here m will be the forwarded carry on b and hence we have another equation m plus b plus b plus b equal to c because there is no forwarded carry on a therefore c must be single digit number and hence we have equation m plus 4 b is equal to c next substitute that value in the equation 3 plus 4 c equal to tan m plus b here c is equal to m plus 4 b if we simplify this equation we have 3 plus 4 m plus 16 b is equal to tan m plus b take b on the left hand side therefore 16 b minus b is equal to 15 b and take 4 m on the right hand side so tan m minus 4 m equal to 6 m now divide this equation by 3 then we have 1 plus since c is only single digit number therefore from this equation m plus 4 b equal to c there is only two possibility for b which is 1 and 2 if b is equal to 3 then we have m plus 3 plus 3 plus 3 plus 3 which is 12 plus m and that means c will be the 2 digit number so it is not possible so only two possibility for b is 1 and 2 suppose b is equal to 2 then 1 plus 5 b will be the odd number and 2 m is the even number so odd number cannot be equal to even number therefore b cannot be 2 and hence b is equal to 1 now use the equation 1 plus 5 b equal to 2 m if you put b equal to 1 that is 1 plus 5 equal to 6 so 2 m equal to 6 therefore m equal to 3 now we have two value m equal to 3 and b equal to 1 substitute in the equation m plus 4 b equal to c then we have 3 plus 4 equal to c therefore c is equal to 7 and hence your final answer is a b c d is equal to 2 1 7 8 such that is 4 times sum will reverse the number and the result is 8 7 1 2 for more such video like share subscribe press the bell icon of next gen mess