Write any four digit number. Repeat the digits to form a eight digit number(eg.-13571357). Show that two of its prime factors are 73 and 137.

Question

Show that two of its prime factors are 73 and 137.

Answer 1

Let the digits of our number be abcd.  Then, we create an 8-digit number by concatenating the digits again:
     abcdabcd
 
The problem asks us to show that two of the prime factors of abcdabcd are 73 and 137.
 
That's easy because abcd*73*137 = abcd*10001 = abcdabcd.
 
The other prime factors of abcdabcd are those prime factors of abcd.

Answer 2

Ishadri,

 

These types of problems (where you repeat some portion of a number) involve factoring out the "repeated portion" and then examining the other factor.

 

So, let's say the number you repeat is represented in digits as ABCD.  We'd have:

 

ABCDABCD = 10,000⋅ABCD + 1⋅ABCD = 10,001⋅ABCD

 

Now, simply note that 10,001 = 73⋅137.

 

Regards,

Hassan