# two teasers about ten-digit numbers

##### Baruch MFE Faculty A 10-digit number A will be written as a_1a_2a_3...a_10. Two separate questions:

1. Suppose a_1 is the number of times the digit '0' appears in A; a_2 is the number of times '1' appears in A, and so on... a_10 is the number of times '9' appears. What is A?

2. (more difficult) A contains each of the digits from 0 to 9 exactly once. The number a_1 is divisible by one. The number a_1a_2 is divisible by two. The number a_1a_2a_3 is divisible by three, and so on...What is A?

#### cgorac

2. (more difficult) A contains each of the digits from 0 to 9 exactly once. The number a_1 is divisible by one. The number a_1a_2 is divisible by two. The number a_1a_2a_3 is divisible by three, and so on...What is A?

3816547290.

• 1. 521010000 ?

#### Tsotne

1. 521010000 ?

Wrong. you have one 5 digit which hasn't been accounted for.

#### Fabrice

Yes, it has since a_6 = 1

#### Sherlock

Wrong. you have one 5 digit which hasn't been accounted for.

I think he meant 5210010000 and he just did a mistake

#### Fabrice

Yes, sorry, forgot an 0

#### Tsotne

Here is the correct answer: 7100000100

2 times a 1

#### Tsotne

2 times a 1

Look at the index view:
0 1 2 3 4 5 6 7 8 9
7 1 0 0 0 0 0 1 0 0

#### Fabrice

yes, but there are two 1 , so a_2 can't be 1

• Tsotne

#### Tsotne

This one: 6210001000

• #### Fabrice

It's the one I proposed, but I misread and thought there were 9 digits

#### Tsotne

I see, yeah now moving to the second problem

#### Tsotne

A 10-digit number A will be written as a_1a_2a_3...a_10. Two separate questions:

2. (more difficult) lol A contains each of the digits from 0 to 9 exactly once. The number a_1 is divisible by one. The number a_1a_2 is divisible by two. The number a_1a_2a_3 is divisible by three, and so on...What is A?

1234567890. Why scratch a head (if I got the problem correctly) or you should have baned this response.

#### Fabrice

is 1234 divisible by 4?

#### Tsotne

is 1234 divisible by 4?

Now I got the problem, I thought about multiplication... 1*2*3*4...

#### Novak

3816547290

firs you figure 0, then 5. then 6-4 combo, then 2, then 8. these are forced. two possibilities for a7: 3 or 7, each of which gives 4 or 6 numbers to check.

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