MATH 140 Class Exercise on Factorization Name__________________________
The goal of this exercise is to find a formula for the function d(n), which is defined by: d(n) equals the number of positive factors of the integer n. For example d(6) = 4 since {1,2,3,6} is the set of all positive factors of 6.
1. Fill in the following chart: for each number n find the number of positive factors of n.
n |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
d(n) |
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n |
10 |
11 |
12 |
13 |
14 |
15 |
16 |
17 |
18 |
d(n) |
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n |
19 |
20 |
21 |
22 |
23 |
24 |
25 |
26 |
27 |
d(n) |
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a) What do you notice about d(n) if n is a prime? ____________________
b) When is d(n)an odd number? __________________________
2. Find d(n) for the following set of numbers given by prime factorization:
n |
20 |
21 |
22 |
23 |
24 |
25 |
d(n) |
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n |
30 |
31 |
32 |
33 |
34 |
35 |
d(n) |
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a) Suppose that n = 2k, what is d(n) = d(2k) = _______________?
b) Does your answer in a) change if the 2 is replaced by 3? __________
c) Find a number n such that d(n) = 10. ____________
3. Find d(n) for the following set of numbers given by prime factorization.
n |
2131 |
2151 |
2171 |
3151 |
3171 |
5171 |
2231 |
d(n) |
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n |
2251 |
2132 |
3251 |
2232 |
2252 |
2331 |
2431 |
d(n) |
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a) Does d(n) depend on the primes or the powers in the factorization? _______________
b) Based on your results what should d(2a3b) be if a and b are integers? ___________
c) Does the same rule seem to hold for primes other than 2 and 3? ________
4. Again, find d(n).
n |
213151 |
213171 |
223151 |
21315171 |
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d(n) |
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Suppose that is the prime
factorization of n where the pi are distinct primes. What is d(n)? _______________