find out a 10 digit number where......?

Question

find out a 10 digit number where in nth place should be a number which denotes number of times n is used in the number. Suppose in 5th place should be a number which denotes number of times 5 is used in the number...

Answer 1

Let's try thinking about it this way. Your number is

bcdefghija in case you take 10th digit as number of 0's

where there are a zeros b , c twos, and so on up to j nines.
Consider what happens when you add all of the digits together. You get

a+b+c+d+e+f+g+h+i+j

but since there are a zeros, b ones, c twos, etc, then this should be
the same thing as

0a+1b+2c+3d+4e+5f+6g+7h+8i+9j,

which gives us an equation:

b+2c+3d+4e+5f+6g+7h+8i+9j = a+b+c+d+e+f+g+h+i+j,

which we can simplify as

c+2d+3e+4f+5g+6h+7i+8j = a

by subtracting the rest of the right hand side from both sides of the
equation. Now a can't be bigger than 9, which means that

g, h, i, j

can't be bigger than 1; in fact, their sum can't be bigger than 1,
which means that either all of them are zero or one of them is 1 and
the rest are all zero. And f can't be bigger than 2. In fact,

f+g+h+i+j

can't be bigger than 2. And

e+f+g+h+i+j

can't be bigger than 3. And

d+e+f+g+h+i+j

can't be bigger than 4. Now try a few cases. What happens if

j = 1 ?

That means that either a=8 and c to i are all zero, or a=9, c=1, and d
to i are all zero. Do either of those work for any number b? If not,
then you have to conclude that j=0. Now what happens if

i = 1 ?

Keep doing this, and you will soon find that you can check all of the
possibilities fairly quickly, and so you can find all possible
solutions.

If you have any questions about this or need more help, please write
back and show me what you have been able to do, and I will try to
offer further suggestions.

Answer 2

Some of this depends on what is meant by "nth place." Does that mean nth place of most significance or nth place of least significance?

If the "1st place" is on the left of the number, then your number is:

1000000000

1 is used once in this number.

2-9 are not used in this number.

10 is not used in this number.

Thus, the first place is a 1 and the 2nd through 10th places are 0's.


If the nth place is the RIGHT of the number, then the number you want is just the reverse:

0000000001


I know that last answer isn't attractive because it has leading zeros; however, the "10th place" must always be a zero because there is no SINGLE digit in base-10 for the number 10, thus the 10th place will always be zero (trivially). That means that you will always have ONE leading zero at least. If that is allowed, then I don't see why 9 leading zeros is any worse.


So in either case, 9 zeros and a 1.

Answer 3

1000000000

Answer 4

1000000000

Answer 5

1000000000
1 is used once.
2 is used 0 times
so is 3,4,5,........,9,10

Answer 6

2100010006