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\n" ); document.write( "If the sum of the odd positive divisors is 24, then the number of those divisors must be even.
\n" ); document.write( "The odd positive integer 1 is a divisor of every integer, so the number of odd positive divisors other than 1 must be odd.
\n" ); document.write( "If there are only 2 odd divisors and the sum of the odd divisors is 24, then those odd divisors must be 1 and 23.
\n" ); document.write( "If there are 4 odd divisors, the largest of them must be the product of the other two. There is only one set of odd divisors that satisfies that condition and has a sum of 24 -- 1, 3, 5, and 15.
\n" ); document.write( "Obviously, the only other prime factors must be even; that means the only other prime factors are 2.
\n" ); document.write( "With the two odd prime factors 1 and 23, any number less than 2022 of the form 23(2^n) satisfies the conditions of the problem. 2022/23 is about 87.9, which is between 2^6=64 and 2^7=128, so the number of factors of 2 is between 0 and 6, inclusive. That gives 7 numbers that satisfy the conditions.
\n" ); document.write( "With the four odd prime factors 1, 3, 5, and 15, any number less than 2022 of the form 15(2^n) satisfies the conditions of the problem. 2022/15 is about 134.8, which is between 2^7=128 and 2^8=256, so the number of factors of 2 is between 0 and 7, inclusive. That gives 8 other numbers that satisfy the conditions.
\n" ); document.write( "ANSWER: the number of positive integers less than 2022 for which the sum of the odd positive divisors is 24 is 7+8 = 15.
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