Reversing a Number

After a pretty boring question yesterday, I found a good one today. Not just good - great. It's a general mathematics question with almost no hidden hints. All it takes to answer that question is pure thinking. Here it is:

There is a number abcde (each letter represents a different digit). When multiplied by 4, this number becomes edcba. What is the number?

Try thinking for a minute. What is the boldest clue in here? Of course, the number of digits. This information alone gives us two digits.

Since there are five digits on both numbers, the first digit, a, must be 1 or 2 (if it's 3 and above, the big number would have six digits). Now, because digit a appears at the units digit of a number divisible by 4, it must be a possible digit of a multiplication by 4. No integer will end in 1 when multiplied by 4, but ending with 2 is possible (for example: 3*4 = 12), so a = 2 and the number is:
2bcde

Now, because this number is multiplied by 4 and starts with 2, it must end with 8 or 9. 4 * 9 = 36, ends with 6, not good. 4 * 8 = 32 - perfect for us, since it ends with 2. So now the number is:
2bcd8

Notice how the bigger number starts with 8 and not 9. This means that b * 4 < 10, so b must be 1 or 2. Since 2 is taken, b = 1:
21cd8

In the bigger number, 8dc12, the tens digit is 1, but 8 * 4 = 32. This means we need a number that when multiplied by 4 gives 8 (to add to the 3). This means d is either 2 or 7. 2 is taken, so d = 7:
21c78

Now two things can be done: one, guessing all possible values of c. Two: calculating c. Since I don't think guessing is a legitimate reason, let's calculate:
c is in the hundreds place, so the number is 21078 + 100c, and the bigger number is 87012 + 100c. That gives the equation:
4(21078 + 100c) = 87012 + 100c
84312 + 400c = 87012 + 100c
300c = 2700
c = 9

Finally, the number is: 21978. Multiply that by 4 and you'll get 87912. Job complete.

Want to see more of these tricks? Join super math tips, there are some really nice things over there.

Hope you liked it,
Nadav

nadavs