Find a 6-digit number abcdef that becomes 5 times as large when the units digit f is moved to the left end of the number. In other words, solve 5 . abcdef = fabcde
Solution:
Let’s consider how a six-digit number changes
when the rightmost digit is moved to the left end. Take 123456 as an example.
Note that 612345 = 600000+12345, whereas 123456 = 123450+6 = 12345 · 10+6.
If we give the name x to 12345, a common technique in algebra, we can write
123456 = 10x + 6 and 612345 = 6 · 10^5 + x. What the hypothesis tells us is that
5(10x+6) = 6·10^5+x.
Of course 123456 does not satisfy the equation, but replacing
abcde with x reduces the six variables to just two.
Of course we don’t know that f = 6 works so we need to solve:
So 5 . abcdef = fabcde can write as:
5 · (10x + f) = f · 10^5 + x
we get 50x + 5f = 10^5f + x
or 49x = (10^5 − 5)f = 99995f
Both sides are multiples of 7 so we can write:
7x = 14285 · f
Now the left side is a multiple of 7, so the right side must also be a
multiple of 7. Since 14285 is not a multiple of 7, that f must be a multiple of 7.
Since f is a digit, it must be 7. And x must be 14285.
Thus: The six digits is
abcdef=142857
Check: 5.142857=714285
True!
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