Re: If integer C is randomly selected from 20 to 99, inclusive.
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11 Feb 2019, 18:18
First work out the givens, we have 20-99 (inclusive) consec ints, so 99-20+1 = 80 ints
Also, C^3-C can be broken down into C(C^2-1) --> (C-1)C(C+1).
We know that any 3 consecutive ints are divisible by 3, so the question really is: How many sets of 3 consec ints from 20 to 99 have enough factors of 2 (12=3*2^2)?
Case 1--> any 3 consec ints with 2 E# starting from 2 are divisible by 4, e.g. {20, 21, 22}
Out of 80 consec ints, half are even. So there are 40 sets of EOE where this is the case.
Case 2--> OEO consec ints when the single even number has at least 2 factors of 2.
e.g. {23,24,25}
So the middle even number needs to be a multiple of 4, Last multiple (96) - First multiple (20) = 76/4 + 1 = 19 + 1 = 20
40+20/80 = 60/80 = 2/3.
The NO Case is how many do NOT have enough factors of 2. These are the even numbers that come up in OEO that are not divisible by 4. e.g. {21, 22, 23}, {25,26,27} ... {97, 98, 99}
We can see that when we divide, 22/4 = 5R2 or 98/4 = 14R2, there's always a remainder of 2, which means that the next or previous even number is divisible by 4.
This might actually be a quicker way to determine the answer.
If half of the 80 is evens and half is odds, that's 40 of each.
We are taking 3 consecutive numbers, so EOE or OEO. When can we not divide the 3 numbers by 4?
When the single even number is not divisible by 4. I.e. Every other even number isn't divisible, so out of 40 OEO half would not be divisible by 4. e.g. 20, 22, 24, 26, 28, 30, 32, 34 ...
So 1/4 of all possible sets of consec ints have only 1 E# number that is not divisible by 4.
We can write out the sequences of 12 numbers where n = 20 through 31 to see the pattern:
n-1, n, n+1
{19,20,21} yes, case 2
{20,21,22} yes, case 1
{21,22,23} no
{22,23,24} yes, case 1
{23,24,25} yes, case 2
{24,25,26} yes, case 1
{25,26,27} no
{26,27,28} yes, case 1
{27,28,29} yes, case 2
{28,29,30} yes, case 1
{29,30,31} no
{30,31,32} yes, case 1
So 9/12=3/4 cases are YES, 3/12=1/4 are NO.
This pattern repeats 4 times completely (from n=20-31, 32-43, 44-55, 56-67, 68-79, 80-91) with 9*6 = 54 YES and 1 time partially (n=92-99) with 6 YES cases, 54+6 = 60 out of 80, which is the same as what we got above.