*Notes to a video lecture on http://www.unizor.com*

__Arithmetic+ 06__

*Problem A*

A positive integer number in decimal notation contains only digits

*and*

**1***.*

**0**The digit

*occurs*

**1***111*times, while the digit

*occurs an unknown number of times.*

**0**Can this number be a square of another integer number?

*Hint A*: Use the rules of divisibility.

*Answer A*: No.

*Problem B*

This problem is based on

*Problems 04(B,C)*of this course

*Math+ & Problems*.

Prove that if the sum of digits of some natural number

*is the same as the sum of digits of the number*

**N***, where*

**k·N**

**k−1****is not divisible by**, then number

**3***is divisible by*

**N***.*

**9***Hint B*:

*Problem 04(B)*stated that a remainder of the division of some natural number by 9 is the same as a remainder of the division by 9 of the sum of this number's digits.

Therefore, both

*and*

**N***have the same remainder if divided by*

**k·N***.*

**9***Problem C*

Given a number

*N*with a decimal representation

*999...9*that contains

*k*digits

*9*.

Assume for definitiveness,

*k*is a prime number.

Find a number whose decimal representation contains only digits

*1*that is divisible by

*N*.

*Answer C*:

*should contain*

**111...1***digits*

**9·k***1*.

*Example C*: For

*(*

**N=99***) the number*

**k=2***that contains*

**111...1***9·2=18*digits

*1*is divisible by

*.*

**N***Problem D*

Consider the number

**N=(k+1)·(k+2)·...·(2k−1)·(2k)**How many

*2*'s, depending on

*, are in the representation of this number as a product of prime numbers?*

**k***Hint D*: Notice that

**N=(2k)!/(k!)**Then

*can be represented as a product of only odd numbers by only even numbers.*

**N=(2k)!***Answer D*:

**N=2**^{k}·Mwhere

*is an odd number,*

**M**so the number of

*2*'s in the representation of number

*as a product of prime numbers is*

**N***.*

**k**
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