Algebra.Com's Answer #812882 by ikleyn(52786)![\"\"](\"/images/stars/stars-13.gif\")  
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document.write( "How many pairs (a,b) of positive integers are there such that \"a\" and \"b\" are factors of 6^6 and \"a\" is a factor of \"b\" ?
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document.write( " In this problem, the terms (the words) \"factor\", \"factors\" are used as synonyms of the term \"divisor\", \r
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document.write( " as it is clear from the context.\r
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document.write( " It means that when the problem says \"a\" and \"b\" are factors of  , it DOES NOT necessary mean that a*b = .\r
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document.write( " According to the context, it ONLY MEANS that \"a\" and \"b\" are the DIVISORS of .\r
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document.write( " It is how I read, understand, interpret and solve this problem.\r
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document.write( "First of all, =  =  .\r\n" );
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document.write( "So, the divisors of the number = are all those and only those integer positive numbers  , \r\n" );
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document.write( "where 0 <= m <= 6, 0 <= n <= 6.\r\n" );
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document.write( "From this notice, it becomes clear that the total number of divisors \" b \" to the number is 7*7 = 49.\r\n" );
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document.write( " | So, we solved part of the problem, but not the whole problem, yet. |\r\n" );
document.write( " | Now I will analyse the remaining part. |\r\n" );
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document.write( "For any factor \"a\" of b =  , the \"a\" has the form a =  , where 0 <= i <= m, 0 <= j <= n.\r\n" );
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document.write( "It implies that for any of 49 factors b = , there are (m+1)*(n+1) factors \"a\" to \"b\".\r\n" );
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document.write( "Hence, when we calculate the number of pairs (a,b), as described in the problem, we sum up all these products (m+1)*(n+1) \r\n" );
document.write( "for all pairs (m,n) with m and n from 0 to 6 inclusive.\r\n" );
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document.write( "From this point, it is not difficult to get the idea that our sough number of pairs (a,b) is THIS PRODUCT\r\n" );
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document.write( " (1 + 2 + 3 + . . . + 6 + 7)*(1 + 2 + 3 + . . . + 6 + 7).\r\n" );
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document.write( "Each factor in parentheses is the sum of the first 7 natural numbers S =  = 7*4 = 28.\r\n" );
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document.write( "Therefore, the answer to the problem's question is \"there are 28*28 = 784 such pairs (a,b)\".\r\n" );
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document.write( "ANSWER. There are 784 such pairs (a,b).\r\n" );
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document.write( "Solved.\r
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document.write( "In my solution, I treat the notion/conception of a divisor to a number N
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document.write( "as any proper divisor AND/OR improper divisors as the number of \" 1 \" and the number N itself.\r
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document.write( "Such treatment makes the solution easier; from the other hand side, it is not prohibited \r
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document.write( "by the text or the context to treat it this way.\r
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