The sum of the digits of a three digit number is 11 If the d

The sum of the digits of a three digit number is 11. If the digits are reversed, the new number is 46 more than five times the old number. If the hundreds digit plus twice the tens digit is equal to the units digit, then what is the number

Solution

Let the three digit number be xyz. Then x + y + z = 11...(1). Also zyx = 5xyz + 46....(2) Further, since the hundreds digit plus twice the tens digit is equal to the units digit, we have x+ 2y = z or, x + 2y - z = 0... (3) On addind the 1st and the 3rd equations, we get 2x + 3y = 11...(4). Assuming x, y, z to be all positive integers, we have only 2 possible solutions for equation (4).The 1st solution is x =1 and y =3; the 2nd solution is x =4 and y = 1.If x = 1 and y = 3, then from the 3rd equation, we have z = 1+ 2*3 = 1+6 = 7 and if x = 4 and y = 1, then from the 3rd equation, z = 4 + 2*1 = 6. Thus, the required number is xyz = 137 or 416. To finally ascertain the correct number, we shall use the 2nd equation. If xyz = 137, then zyx = 731. Also 5xyz + 46 = 5*137 + 46 = 685+ 46 = 731 = zyx. Thus 137 is a correct solution. Further, if xyz = 416, then zyx = 614 and 5 * 416 + 46 = 2080 + 46 = 2126 614. Thus 137 is the required number.