I have always been interested in ways and techniques people use to solve mathematical problems possibly because I have always struggled with math. Recently I read a wonderful article which was describing a property of perfect numbers i.e. natural numbers whose positive divisors including 1 but excluding itself sum to the number, for example 6 is a perfect number because the divisors of 6 (1,2,3,6), including 1 but excluding 6 sum to 6 (6 = 1 + 2 + 3).
The author was explaining that the prime number representation of a natural number n can be used to find the sum of the principal divisors. For example, if n = p1^e1 * p2^e2 * ...*pn^en then the sum of all the divisors is given by (1 + p1 + p1^2 ... + p1^e1) x (1 + p2 + p2^2 + ... + p2^e2) x...x(1+ pn + pn^2 + ... + pn^en). And since each of the terms is a geometric series it follows that this product is reduced to ( (p1^(e1+1) - 1)/(e1 - 1) ) X ( (p2^(e2+1) - 1)/(e2 - 1) ) X ... X ( (pn^(en+1) - 1)/(en - 1) ). However this left me perplexed how someone could think of multiplication of such nice series?
In order to investigate this issue let us consider a simple example. Let the natural number n = 12. Now, 12 = 2.2.3 or 2^2 . 3. According to the previous expression sum of the divisors can be written as (1 + 2^1 + 2^2) X (1 + 3) = ((2^(2+1) - 1)/(2-1) ) X (((3^(1+1) - 1)/(3-1)) = ((2^3 - 1)/1) X ((3^2 - 1)/2) = 7 . 4 = 28. On the other hand the positive divisors of 12 are {1,2,3,4,6,12}. Observe that the sum that we calculated include the number (here 12) as well i.e. 1+2+3+4+6+12 = 28.
Our problem can also be framed as start with the expression 1 + 2 + 3 + 4 + 6 + 12 and deduce (1 + 2 + 2^2) . (1 + 3). Observe that 1 + 2 + 3 + 4 + 6 + 12 = 1 + 2 + 3 + 2.2 + 2.3 + 2.2.3 = 1.1 + 2.1 + 1.3 + 2^2.1 + 2.3 + 2^2 . 3 = 1.(1 + 3) + 2.( 1 + 3) + 2^2(1 + 3) = (1 + 2 + 2^2).( 1 + 3) . If we multiply out the expression (1 + 2 + 2^2) ( 1 + 3), now it seems clear that all terms will divide 12.
Although the above stated problem/example is trivial but it does teach at least one problem solving technique - generally the expressions are associated with some kind of "fill in the blank" and as soon as the blank is filled the way to deduce other things might become obvious. For example in the previous example things become clear when we expressed all single digit terms as the product of the digit and 1 i.e. 2 = 1.2.
The author was explaining that the prime number representation of a natural number n can be used to find the sum of the principal divisors. For example, if n = p1^e1 * p2^e2 * ...*pn^en then the sum of all the divisors is given by (1 + p1 + p1^2 ... + p1^e1) x (1 + p2 + p2^2 + ... + p2^e2) x...x(1+ pn + pn^2 + ... + pn^en). And since each of the terms is a geometric series it follows that this product is reduced to ( (p1^(e1+1) - 1)/(e1 - 1) ) X ( (p2^(e2+1) - 1)/(e2 - 1) ) X ... X ( (pn^(en+1) - 1)/(en - 1) ). However this left me perplexed how someone could think of multiplication of such nice series?
In order to investigate this issue let us consider a simple example. Let the natural number n = 12. Now, 12 = 2.2.3 or 2^2 . 3. According to the previous expression sum of the divisors can be written as (1 + 2^1 + 2^2) X (1 + 3) = ((2^(2+1) - 1)/(2-1) ) X (((3^(1+1) - 1)/(3-1)) = ((2^3 - 1)/1) X ((3^2 - 1)/2) = 7 . 4 = 28. On the other hand the positive divisors of 12 are {1,2,3,4,6,12}. Observe that the sum that we calculated include the number (here 12) as well i.e. 1+2+3+4+6+12 = 28.
Our problem can also be framed as start with the expression 1 + 2 + 3 + 4 + 6 + 12 and deduce (1 + 2 + 2^2) . (1 + 3). Observe that 1 + 2 + 3 + 4 + 6 + 12 = 1 + 2 + 3 + 2.2 + 2.3 + 2.2.3 = 1.1 + 2.1 + 1.3 + 2^2.1 + 2.3 + 2^2 . 3 = 1.(1 + 3) + 2.( 1 + 3) + 2^2(1 + 3) = (1 + 2 + 2^2).( 1 + 3) . If we multiply out the expression (1 + 2 + 2^2) ( 1 + 3), now it seems clear that all terms will divide 12.
Although the above stated problem/example is trivial but it does teach at least one problem solving technique - generally the expressions are associated with some kind of "fill in the blank" and as soon as the blank is filled the way to deduce other things might become obvious. For example in the previous example things become clear when we expressed all single digit terms as the product of the digit and 1 i.e. 2 = 1.2.
1 comment:
You have made a good observations of the number.
Post a Comment