I would appreciate if you could help me with this problem.

Say you have a four digit number (e.g. abcd) and you multiply the number by 4. The answer you get will be the reverse order of the number you started with (dcba). What is the number? The four digits (a,b,c,d) cannot be the same number or cannot be repeated.

Thanks for you help.


I can get you started. The digit a can't be very large if abcd 4 is a 4 digit number. In fact 4  3 = 12 so if a = 3 then abcd  4 is at least five digits. Thus a is either 1 or 2. If a is 1 then 1bcd  4 = dcb1 But this is not possible since the left side is even and the right side is odd. Thus a = 2. Hence

2bcd  4 = dcb2

Look at d on the right side. What can it be? 4  2 = 8 so d might be 8 or, if there is a carry, d might be 9. But if d on the left side is 9 then 4  9 = 36 which conflicts with the units digit of 2 on the right side. Thus d = 8 and we have

2bc8  4 = 8cb2

Can you finish it now?

What about 5 digits. Can you find a five digit number abcde so that abcde  4 = edcba?

Penny and Claude

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