Riddles, brain puzzles and mathematical problems

[MyRedNeptune]

Give me the number and explain your method. You don't have anything to lose.

The number is supposed to be big (really big)

Spoiler below!

Let's suppose that a is equal to the first digit of n and b is equal to the number after the first digit of n. Now we can write a simple equation for one potential value of n:

n = 2m

10a + b = 2( 10b + a)
10a + b = 20b + 2a
19b = 8a

1 <= a <= 9 , so in this case we can definitely see that the equation has no solution. Let's look at some other variants:

100a + b = 2( 10b + a)
19b = 98a

1000a + b = 2(10b + a)
19b = 998a

10000a + b = 2(10b + a)
19b = 9998a

We can see that the coefficient of a changes with the assumed length of n. Finding the right coefficient will allow for a valid solution to be found. Let's label the coefficient c. Because c follows a particular pattern of growth, we can notice a pattern in its multiples with a:

998 * 2 = 1996
9998 * 2 = 19996
9998 * 4 = 39992
9998 * 8 = 79984
99998 * 8 = 799984

We see that the value of a determines the first digit and the last two digits of a*c, while the value of c affects the amount of 9's between them. Let's label the first digit of a*c as x and the last two digits of a*c as y. The table below shows the values of x and y that correspond to all possible values of a:

Code: (Select All)

a     x     y
1     0     98
2     1     96
3     2     94
4     3     92
5     4     90
6     5     88
7     6     86
8     7     84
9     8     82

Because 19b = a*c , we know that a*c must be divisible by 19. Now we can pick any row from the table above and start progressively adding 9's between x and y until the resulting number can be divided by 19. Then we have found a*c, from which b, n and m can be found.

For example:

19b = 099999999999999998
b = 05263157894736842
a = 1
n = 105263157894736842
m = 52631578947368421

19b = 199999999999999996
b = 10526315789473684
a = 2
n = 210526315789473684
m = 105263157894736842