7digit no if you add 1to it it will become 9 digit no

Question

It is a riddle n it is given as a home work

Answer 1

One solution is to consider different number bases. Adding 1 to a 7-digit number in one base equals a 9-digit number in a different base. If we call the bases a and b (b<a) we can represent a 7-digit number in one base (base a) as a range of integers a⁶ to a⁷-1, and a 9-digit number in the other base as b⁸ to b⁹-1.

We should be able to find at least one number x such that x_b=x_a+1. 

The lowest base is 2 and the next lowest is 3, so b=2 and a=3. To find out if these bases can be used to find a solution, we need to look at 3⁶ and 3⁷-1, that is, 729₁₀ to 2186₁₀; and 2⁸ and 2⁹-1, that is, 256₁₀ to 511₁₀.

These two ranges don't overlap, so we can't use these bases.

Let's move to a=4 and b=3. This time we have 4⁶ and 4⁷-1, a range 4096₁₀ to 16383₁₀; and 3⁸ and 3⁹-1, b range 6561₁₀ to 19682₁₀. These ranges overlap between 6561₁₀ and 16383₁₀, so x can be any number in the range 6561≤x≤16383. If we take the central number x = (6561+16383)/2=11472, we need to convert this number into the bases: 11472=2303100₄ (7 digits) and 120201220₃ (9 digits).

Now add 1 to the 7-digit number: 2303101₄ and this becomes the 9-digit number: 120201221₃.

2303101₄=2×4096+3×1024+3×64+16+1=8192+3072+192+16+1=11473₁₀.

120201221₃=6561+2×2187+2×243+27+2×9+2×3+1=6561+4374+486+27+18+6+1=11473₁₀.