**#Problem 4**

How many primes among the positive integers, written as usual in base 10, are such that their digits are altenating 1’s and 0’s, beginning and ending with 1?

**[Putnam]**

**Solution scheme and approach**

Let,

X=1010101…1=1+10^2+10^4+…+10^2n=(10^2n-1)/99=(10^n+1)*(10^n-1)/99

For n=2 X=101 which is a prime number

For n>=3

If n is even

{

As (10^n+1)>99

(10^n-1)=(9+1)^n -1=0(mod 9) So,(10^n-1)=9a

(10^n-1)=(11-1)^n -1=0(mod 11)So (10^n-1)=11b

therefore we can write X=99*a*b*(10^n+1)=>Composite

}

If n is odd

{

As (10^n-1)>99

(10^n-1)=(9+1)^n -1=0(mod 9) So, (10^n-1)=9a[a>1]

(10^n+1)=(11-1)^n +1=0(mod 11) So,(10^n+1)=11b[b>1]

therefore X=99*a*b=>composite

}

**So, there is only one prime number, 101, exists.Q.E.D.**

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