Number magic - math puzzle
If you multiply me by 2, subtract 1, and read the reverse the result you'll find me. Which numbers can I be?
Explanation
37, 397, 3997, 39997, 399.....997 etc.
Because
37 * 2 - 1 = 73
397 * 2 - 1 = 793
399....997 * 2 - 1 = 799....993
To find these numbers, we first investigate the possible number which consist of two digits. The number x we want to find we write as x = AB, where A and B are two digits. In other words, x = 10 * A + B.
It holds that 2 * x - 1 = 20*A + 2*B - 1. If you reverse this number, you have to get the original number AB back. But then it of course also hold that if you reverse the original number (that becomes 10 * B + A), you will get 20*A + 2*B - 1. Hence
20 * A + 2 * B - 1 = 10 * B + A
We can simplify this equation to
19 * A = 8 * B + 1.
The right part of this equation is always odd. So the left part should be odd as well. This implies that A should be odd. The right part is always smaller than 73. So A cannot be larger than 3. If we try A = 1, we do not find a solution. But A = 3 has a solution with B = 7. So the number we search for is 37. This is the only number with two digits that satisfies our requirements.
Because
37 * 2 - 1 = 73
397 * 2 - 1 = 793
399....997 * 2 - 1 = 799....993
To find these numbers, we first investigate the possible number which consist of two digits. The number x we want to find we write as x = AB, where A and B are two digits. In other words, x = 10 * A + B.
It holds that 2 * x - 1 = 20*A + 2*B - 1. If you reverse this number, you have to get the original number AB back. But then it of course also hold that if you reverse the original number (that becomes 10 * B + A), you will get 20*A + 2*B - 1. Hence
20 * A + 2 * B - 1 = 10 * B + A
We can simplify this equation to
19 * A = 8 * B + 1.
The right part of this equation is always odd. So the left part should be odd as well. This implies that A should be odd. The right part is always smaller than 73. So A cannot be larger than 3. If we try A = 1, we do not find a solution. But A = 3 has a solution with B = 7. So the number we search for is 37. This is the only number with two digits that satisfies our requirements.
|
|