document.write( "Question 1138467: Determine the number of ordered pairs of positive integers (a,b) such that a and b are both divisors of 1260 and a/b \n" ); document.write( "

Algebra.Com's Answer #756297 by greenestamps(13200) $\"\"$ $\"About$

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\n" ); document.write( "The prime factorization of 1260 is 1260 = 2*2*3*3*5*7

\n" ); document.write( "One of the factors of 1260 is 20 = 2*2*5; 1260/20 = 63.

\n" ); document.write( "If a=20, then b can be any number of the form 20n, where n is any factor of 63.

\n" ); document.write( "63 = 3*3*7; the number of factors of 63 is (2+1)(1+1) = 3*2 = 6.

\n" ); document.write( "So there are 6 ordered pairs of positive integers (20,b) for which 20 and b are both divisors of 1260 and a divides b: (20,20), (20,60), (20,140), (20,180), (20,420), and (20,1260).

\n" ); document.write( "Perform a similar analysis for each value of a:\"

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document.write( "    +-------------------------------------------+\r\n" );
document.write( "    |   |          |             |  number of   |\r\n" );
document.write( "    |   |          |             |  possible    |\r\n" );
document.write( "    |   |          | factors     |   values     |\r\n" );
document.write( "    | a | n=1260/a |   of n      |   for b      |\r\n" );
document.write( "    +-------------------------------------------+\r\n" );
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document.write( "      1     1260     2*2*3*3*5*7   3*3*2*2 = 36\r\n" );
document.write( "      2      630     2*3*3*5*7     2*3*2*2 = 24\r\n" );
document.write( "      3      420     2*2*3*5*7     3*2*2*2 = 24\r\n" );
document.write( "      4      315     3*3*5*7       3*2*2   = 12\r\n" );
document.write( "      5      252     2*2*3*3*7     3*3*2   = 18\r\n" );
document.write( "      6      210     2*3*5*7       2*2*2*2 = 16\r\n" );
document.write( "      7      180     2*2*3*3*5     3*3*2   = 18\r\n" );
document.write( "      9      140     2*2*5*7       3*2*2   = 18\r\n" );
document.write( "     10      126     2*3*3*7       2*3*2   = 12\r\n" );
document.write( "     12      105     3*5*7         2*2*2   =  8\r\n" );
document.write( "     14       90     2*3*3*5       2*3*2   = 12\r\n" );
document.write( "     15       84     2*2*3*7       3*2*2   = 12\r\n" );
document.write( "     18       70     2*5*7         2*2*2   =  8\r\n" );
document.write( "     20       63     3*3*7         3*2     =  6\r\n" );
document.write( "     21       60     2*2*3*5       3*2*2   = 12\r\n" );
document.write( "     28       45     3*3*5         3*2     =  6\r\n" );
document.write( "     30       42     2*3*7         2*2*2   =  8\r\n" );
document.write( "     35       36     2*2*3*3       3*3     =  9\r\n" );
document.write( "     36       35     5*7           2*2     =  4\r\n" );
document.write( "     42       30     2*3*5         2*2*2   =  8\r\n" );
document.write( "     45       28     2*2*7         3*2     =  6\r\n" );
document.write( "     60       21     3*7           2*2     =  4\r\n" );
document.write( "     63       20     2*2*5         3*2     =  6\r\n" );
document.write( "     70       18     2*3*3         2*3     =  6\r\n" );
document.write( "     84       15     3*5           2*2     =  4\r\n" );
document.write( "     90       14     2*7           2*2     =  4\r\n" );
document.write( "    105       12     2*2*3         3*2     =  6\r\n" );
document.write( "    126       10     2*5           2*2     =  4\r\n" );
document.write( "    140        9     3*3           3       =  3\r\n" );
document.write( "    180        7     7             2       =  2\r\n" );
document.write( "    210        6     2*3           2*2     =  4\r\n" );
document.write( "    252        5     5             2       =  2\r\n" );
document.write( "    315        4     2*2           3       =  3\r\n" );
document.write( "    420        3     3             2       =  2\r\n" );
document.write( "    630        2     2             2       =  2\r\n" );
document.write( "   1260        1                           =  1\r\n" );
document.write( "                             ------------------\r\n" );
document.write( "                                total:      330

\n" ); document.write( "ANSWER: There are 330 ordered pairs (a,b) in which a and b are both divisors of 1260 and a divides b.\r
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