The document discusses finding the smallest number with exactly 12 factors. It explains that a number with prime factors of the form Pa1Pb2Pc3 will have (a+1)(b+1)(c+1) factors. For a number to have 12 factors, the expression (a+1)(b+1)(c+1) must equal 12. It determines that the smallest such number is 60, since it has prime factors of 2, 2, 3 and 5, satisfying the expression.

1. Number Theory

2. Number Theory
What is the smallest number that has exactly 12 factors?
(a) 211 (b) 60
(c) 120 (d) 144

3. Number Theory
Solution:
A number of the form P1
aP2
bP3
c will have (a + 1)(b + 1)(c + 1) factors
where P1, P2, P3 are prime.
If a number has exactly 12 factors then (a + 1)(b + 1)(c + 1) should be
equal to 12.
What is the smallest number that has exactly 12 factors?

4. Number Theory
If the number has only one prime factor, it should be of the form.
P11 smallest no possible = 211
If it has 2 prime factors, it can P1
3P2
2 or P1
5P2
23 x 32 or 25 x 3
72 or 96
What is the smallest number that has exactly 12 factors?

5. Number Theory
If it has 3 prime factors it can be of the form
P1
aP2
bP3
c
P1P2P3 = 22 x 3 x 5
P1P2P3 = 60
The smallest number possible = 60
Answer Choice (B)
What is the smallest number that has exactly 12 factors?

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