document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1170032>Question 1170032</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The integers a, b, c, and d are each equal to one of 1, 2, 3, 4, 5, 6, 7, 8, 9. (It is<BR>\n" );
document.write( "possible that two or more of the integers a, b, c, and d have the same value.) The<BR>\n" );
document.write( "integer P equals the product of a, b, c, and d; that is, P = abcd.<BR>\n" );
document.write( "(a) Determine whether or not P can be a multiple of 216.<BR>\n" );
document.write( "(b) Determine whether or not P can be a multiple of 2000.<BR>\n" );
document.write( "(c) Determine the number of possible different values of P that are divisible by 128<BR>\n" );
document.write( "but are not divisible by 1024.<BR>\n" );
document.write( "(d) Determine the number of ordered quadruples (a, b, c, d) for which P is 98 less<BR>\n" );
document.write( "than a multiple of 100. </SPAN>\n" );
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document.write( "(a) multiple of 216?<br><BR>\n" );
document.write( "216 = (2^3)(3^3)<br><BR>\n" );
document.write( "In choosing 4 (not necessarily different) integers from 1 to 9, we need at least 3 factors of 2 and at least 3 factors of 3.<BR>\n" );
document.write( "9*8*3 gives us 3 factors of 2 and 3 factors of 3; so (for example) we could choose those 3 numbers and any of the 9 numbers for the fourth.<br><BR>\n" );
document.write( "ANSWER: Yes, P can be a multiple of 216.<br><BR>\n" );
document.write( "(b) multiple of 2000?<br><BR>\n" );
document.write( "2000 = (2^4)(5^3)<br><BR>\n" );
document.write( "Among the integers 1 through 5, there is only one factor of 5.  So we would need to choose 5 for 3 of the 4 numbers.  But then the fourth number would have to contain 4 factors of 2, which it can't.<br><BR>\n" );
document.write( "ANSWER: No, P can't be a multiple of 2000.<br><BR>\n" );
document.write( "(c) number of values of P that are multiples of 128 but not of 1024?<br><BR>\n" );
document.write( "128 = 2^7; 1024 = 2^10<br><BR>\n" );
document.write( "We are looking for combinations of 4 of the integers from 1 to 9 that together contain at least 7 factors of 2 but not 10.<br><BR>\n" );
document.write( "To get 7 factors of 2, we need to use all the even numbers, 2, 4, 6, and 8; together they have exactly 7 factors of 2.<br><BR>\n" );
document.write( "Since we had to use 4 of the integers to get the required 7 factors of 2, there is only the one way to get a number divisible by 128: 2*4*6*8 = 384 (= 3*128).<br><BR>\n" );
document.write( "ANSWER: Only one value of P divisible by 128; and it is not divisible by 1024.<br><BR>\n" );
document.write( "(d) number of ORDERED quadruples for which P is 98 less than a multiple of 100 (2, 102, 202, 302, ...)<br><BR>\n" );
document.write( "This one requires a lot of investigation, since the product can be any of a large number of numbers. However, use of a prime factorization calculator shows that the number 2 is the only number in the list that can be written as the product of 4 integers from 1 through 9.<br><BR>\n" );
document.write( "2 = 1*1*1*2<br><BR>\n" );
document.write( "Since this question asks for the number of ORDERED quadruples, the answer is 4 (the single 2 can be any of the four numbers).<br><BR>\n" );
document.write( "ANSWER: 4 ordered quadruples with P equal to 98 less than a multiple of 100:<BR>\n" );
document.write( "1*1*1*2<BR>\n" );
document.write( "1*1*2*1<BR>\n" );
document.write( "1*2*1*1<BR>\n" );
document.write( "2*1*1*1<br><BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );