Bosco

Let P (n) denote the number of factors of n. Find all values of  n which satisfy P(n) = n.    

n =   1     1 factor:    1    

n =   2     2 factors:  1, 2    

n =   3     2 factors    

n =   4     3 factors:  1, 2, 4    

n =   5     2 factors    

n =   6     4 factors:  1, 2, 3, 6    

n =   7     2 factors    

n =   8     4 factors:  1, 2, 4, 8    

n =   9     2 factors    

n = 10     4 factors:  1, 2, 5, 10    

n = 11     2 factors   

n = 12     6 factors:  1, 2, 3, 4, 6, 12    

Those two are all I can think of.  As n gets larger, P(n) sometimes gets larger,     

but I think the trending demonstrates that P(n) is never going to catch up to n.    

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How many positive integers are there whose digits strictly decrease from left to right,     

and have at most one even digit, and the sum of the digits is 6?    

 6 0          This is legit, because zero isn't even.  Somebody might argue definitions.    

 5 1    

 5 1 0    

 4 2           Won't work, problem says there can't be more than one even digit.    

 3 2 1    

 3 2 1 0    

By the terms of the problem, I find 5 of them.    

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Find the area of the triangle with side lengths 4, 5, and 8.    

I've always been sort of partial to Heron's formula.    

Area = sqrt[(s)(s–a)(s–b)(s–c)]    where s = (1/2)(a+b+c)    

s = (1/2)(17) = 8.5     therefore A = sqrt(8.5 x 4.5 x 3.5 x 0.5)    

                                                 A = sqrt(66.9375)    

                                                 A = 8.1815    

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Find the positive difference between the solutions to the equation 6t^2 + 30 = 41t    

                                     6t² + 30  =  41t    

                                     6t² – 41t + 30  =  0    

this will factor              (6t – 5)(t – 6)  =  0    

                                       t – 6 = 0     ——>     t  =  6    

                                     6t – 5 = 0     ——>     t  =  ⁵/₆    

difference between                                             5 ¹/₆    

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Find the area of the region enclosed by the graph of x^2 + y^2 = 2x - 6y + 6 + 4x - 8y.    

                                      x² + y²  =  2x – 6y + 6 + 4x – 8y    

rearrange & group         (x² – 6x    ) + (y² + 14y    )  =  6    

Complete the squares on x and y.  Add the same amount to the right side that you add to the left.    

                                           (x² – 6x + 9) + (y² + 14y + 49)  =  6 + 9 + 49    

                                           (x – 3)² + (y + 7)²  =  64    

That's a circle in the form   (x – h)² + (y –k)² = r²  where h and k locate the center and r is the radius.    

Area of the circle is A = π r²     ——>>  A = 64 π  =  201.06    

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A committee is to be made up from a group of eight candidates. The committee must consist of at least two people, but at most four people. How many ways can the committee be chosen?    

₈C₂  =  28    

₈C₃  =  56    

₈C₄  =  70    

Add up the possibilities.    

The total comes to 154    

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The area of Kaitlyn's square garden is 360 ft2. One side of the garden is next to a shed. She wants to put a fence around the other three sides of the garden. Determine the approximate amount of fence it will take.    

sqrt(360)  =  approx  19     the length of one side of the garden    

      3 x 19  =  57 ft    

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Given that a^2-b^2=704 and a+b=44, find the value of a-b.    

Write down the first eq                   a² – b²  =  704    

Factor it to                             (a + b)(a – b)  =  704     (eq 1)    

Write down the other eq        (a + b)            =    44     (eq 2)    

See where this is going?    

Divide the left   part of eq 2 into the   left part of eq 1    

Divide the right part of eq 2 into the right part of eq 2    

This gives us (a – b) on the left and 16 on the right    

They're equal, so a – b = 16    

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How many sides has an equiangular polygon if the sum of its exterior angles is equal to the sum of its interior angles?​    

The sum of the exterior angles of any regular polygon is 360^o      

Therefore the sum of the interior angles of our polygon is 360^o    

That makes it a square, which has four sides    

I just recognized it when I saw it, but for a difficult problem, we could have used the formula.   

The sum of the interior angles of a polygon with 'n' sides is given by the formula: S = (n – 2) • 180^o.    

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