P is a positive decimal (base 10) integer consisting entirely of the digit 3, and Q is a positive decimal integer consisting entirely of the digit 7. In the base-10 expansion of P*Q, the digit 3 is repeated precisely three times and the digit 7 is repeated precisely seven times. The product P*Q may consist of other digits besides 3 and 7.

Given that N is the minimum value of P*Q, determine the remainder when N is divided by 37.

Note: Try to derive a non computer assisted method, although computer programs/spreadsheet solutions are welcome.

I saw this just the evening before it arrived here (in another location on this site).

I was fascinated by the series, for want of a better word, when P, as a number of 3's is multiplied by Q which is also a number of Q's:

        3 * 7 =     21       3 *         7 =       21
      33 * 7 =    231       3 *       77 =     231
    333 * 7 =  2331       3 *      777 =   2331
  3333 * 7 = 23331       3 *    7777 = 23331

But then,
       33  *      77  =               2541
     333  *     777  =           258741
  33333  * 77777  =    2592540741

Soon after this MS Excel refuses to generate true products, it looses precision.

I assume that there is a lovely property that might allow me to determine where lie the respective 3's and 7's in the appropriate length of P*Q.

That 37 is listed as a divisor seems somewhat whimsical too.  Is there some coincidence of 3's and 7's there too?

Since the spreadsheet seems to offer little joy I doubt that a computer listing is readily available without more in depth of the extent of that "lovely property" noted above.

Posted by brianjn on 2008-11-02 23:35:38