Prove the following about perfect numbers:
1. Any odd perfect number can't be a square number.
2. The sum of reciprocals of factors of a perfect number is equal to that perfect number.
3. Prove that odd perfect numbers are of the form:
N=p^(4x+1)* Q^2, where p is of the form (4y+1).
4. Also prove that no odd perfect number is a multiple of 105.
5. The smallest factor of an odd perfect number <= (2k + 6)/3, where k is the number of it's prime factors.
6. Prove that, if p is of the form (12k + 1), corresponding odd perfect number will always be of the form (12k + 1).
Monday, July 25, 2011
Tuesday, July 19, 2011
Math problems for High-school kids
Here are some interesting math problems for high-school grade.
1. When written in decimal notation, how many trailing zeroes will be there in 32! In general, what is the number of zeroes for n! (any positive n).
2. How many numbers from 1 to 100 (inclusive) have odd number of factors? What type of numbers are those? Why? (Factors include one and that number as well).
3. Which number(s) from 100 to 200 (inclusive) has the highest number of factors? How many does it have? (It is better not to use brute-force, try to use "some" method).
4. Prove that the number of factors of a number n, is always not greater than 2 * floor(sqrt(n)).
5. What is the biggest number n, which has number of factors equal to 2 * floor(sqrt(n))?
6. Prove that, x^4n + y^4n = z^4n is not possible whenever x mod 6 = 1 or -1 (5), and y mod 6 = 3, and z mod 6 = 2 or -2 (4)?
7. Prove that the only possible cases for x, y and z above are:
x mod 6 = 0, y mod 6 = 1 or -1, z mod 6 = 1 or -1.
x mod 6 = 3, y mod 6 = 2 or -2, z mod 6 = 1 or -1.
This reduces the cases for the theorem (fermat's last theorem's special case), from a possible 36 combinations to a mere 4.
8. Prove that the powers of 4, always end with either 4 or 6 (in decimal notation). With what digit does 3^1000 end? What number do powers of 6, end with? What about powers of 5?
9. Can you extend the above observation to two/three digit numbers and beyond? For example, with what digits do powers of 3456 end?
10. Prove that for any number a (greater than 2), a^n - 1 (n greater than 1), is a composite number always. Also prove that a^(2*n+1) + 1 is always a composite number?
11. Prove that in x^4 + y^4 = z^4, at least one of x or y is divisible by 5. (If you take the irreducible forms of x and y, then it follows that exactly one of them is divisible by 5).
1. When written in decimal notation, how many trailing zeroes will be there in 32! In general, what is the number of zeroes for n! (any positive n).
2. How many numbers from 1 to 100 (inclusive) have odd number of factors? What type of numbers are those? Why? (Factors include one and that number as well).
3. Which number(s) from 100 to 200 (inclusive) has the highest number of factors? How many does it have? (It is better not to use brute-force, try to use "some" method).
4. Prove that the number of factors of a number n, is always not greater than 2 * floor(sqrt(n)).
5. What is the biggest number n, which has number of factors equal to 2 * floor(sqrt(n))?
6. Prove that, x^4n + y^4n = z^4n is not possible whenever x mod 6 = 1 or -1 (5), and y mod 6 = 3, and z mod 6 = 2 or -2 (4)?
7. Prove that the only possible cases for x, y and z above are:
x mod 6 = 0, y mod 6 = 1 or -1, z mod 6 = 1 or -1.
x mod 6 = 3, y mod 6 = 2 or -2, z mod 6 = 1 or -1.
This reduces the cases for the theorem (fermat's last theorem's special case), from a possible 36 combinations to a mere 4.
8. Prove that the powers of 4, always end with either 4 or 6 (in decimal notation). With what digit does 3^1000 end? What number do powers of 6, end with? What about powers of 5?
9. Can you extend the above observation to two/three digit numbers and beyond? For example, with what digits do powers of 3456 end?
10. Prove that for any number a (greater than 2), a^n - 1 (n greater than 1), is a composite number always. Also prove that a^(2*n+1) + 1 is always a composite number?
11. Prove that in x^4 + y^4 = z^4, at least one of x or y is divisible by 5. (If you take the irreducible forms of x and y, then it follows that exactly one of them is divisible by 5).
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