Finding integers!?Show that 5n+10 gives the correct slope for line tangent to circle with radius 13 @ (-5,12)?

Question

1)Find an integer, x, greater than 4 where the sum of x consecutive integers is divisible by x

2) Show that the equation above gives the correct slope for the line tangent to a circle with a radius of 13 at the point (-5,12)

So I'm thinking, the circle's center is at (0,0) because then the point of tangency will be (-5,12) exactly...But I don't know what to do

For #1, I got 5n+10....is that right or wrong

Answer 1

Consecutive integers can be written:

n, n+1, n+2, n+3, n+4

= 5n + 10 which is divisible by 5

if we try some more

sums will be

6n + 15 (not div. by 6)

7n + 21 (IS div by 7)

8n + 28 (no)

9n + 36 (yes)

10n + 45 (no)

11n + 55 (yes)

It seems to work whenever n is odd.

Sum of first (n-1) integers is

n (n-1) / 2

either n or n-1 is even,

if n-1 is even, then the number is divisible by n

(because n-1 absorbed the factor of 2)

and that occurs when n is odd.

Another way to look at it:

consecutive integers are

(n - k) (n - k + 1) ... (n-1) n (n + 1) ... (n + j -1 ) (n + j)

if k = j, then all the pluses and minuses cancel out

and so you then have (2k+1) n as the sum,

which is divisible by (2k+1), which is any odd number.

k on one side and k on the other + 1 in the middle.

For the circle problem:

The circle is centered at 0,0.

The radius to -5,12 has slope -12/5

so tangent line will have slope 5/12 (perpendicular to radius)

and include that point

y = 5/12 x + b

substitute -5, 12

12 = 5/12 (-5) + b

144 = -25 + 12 b

b = 169/12

Line is

y = 5/12 x + 169/12

=