Find a six-digit number containing no zeros and no repeated digits that satisfies the following conditions:
1. The first and fourth digits sum to the last digit, as do the third and fifth digits. 2. The first and second digits when read as a two-digit number equal one quarter the fourth and fifth digits. 3. The last digit is four times the third digit.
If you call the number ABCDEF, then you get the following equations.
1. A + D = F and C + E = F 2. AB = DE / 4 3. F = 4 × C
The only numbers that work for C and E are 2 and 6 or 4 and 8, and in order to make F a single-digit number, we can deduce that C = 2, E = 6 and F = 8.
So far, our number is AB2D68.
We know A + D = 8 so A and D are both odd numbers. The only odd number less than 8 that we can use for D to make one-quarter of two-digit number D6 also be a two-digit number is 7, so D = 7 and A is 1. This makes the two-digit number AB 19.