Question

If A, B, C, D, E, and F represent different digits, find their values such that: A B C D E F G H × H 1 1 1 1 1 1 1 1 1

Answer 1

The answer is "
`142,857 * 6=857,142`"

Explanation:

Please find the complete question in the attached file.

Let:

`a b c d e f * 6 = d e f a b c`

`\to (10^5 a+10^4 b+10^3c+10^2d+10e+f) * 6 =10^5d+10^4e+10^3f+10^2a+10b+c \\\\\to ( 6 * 10^5 a+ 6 * 10^4 b+6 * 10^3c+ 6 * 10^2d+ 6 * 10e+ 6 * f) =10^5d+10^4e+10^3f+10^2a+10b+c \\\\\to 600,000a+60,000b+6,000c+600d+60e+6f=100,000d+10,000e+1,000f+100a+10b+c \\\\\to 599,900a+59,990b+5999c=99,400d+9,940e+994f\\\\`

`\to 7(599,900a+59,990b+5999c)=7(99,400d+9,940e+994f)\\\\\to 85,700a+8,570b+857c=14,200d+1420e+142f\\\\\to 857(100a+10b+c)=142(100d+10e+f)\\\\\to ((100a+10b+c))/((100d+10e+f))=(142)/(857)\\\\\to (abc)/(142)=(de f)/(857)\\`

therefore 142 and 857 are mutually exclusive, so, the only possible way is:

`\to a=1; b=4; c=2; d=8; e=5; f=7 \\\\ \therefore \\\\ \to 142,857 * 6=857,142`