Here's the 2nd part where there's more math questions I don't get..
-A number is "nifty" if it is a multiple of 2 or 3. How many nifty numbers are there between -11 and 11?
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If for any positive integer n, ┬(n) represents the number of positive divisors of n, what is the value of ┬(┬(┬(12)))?
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Answers and thorough explanations are greatly appreciated.
(especially the 2nd one. I have no clue about the 2nd one...)
Thanks!
-A number is "nifty" if it is a multiple of 2 or 3. How many nifty numbers are there between -11 and 11?
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If for any positive integer n, ┬(n) represents the number of positive divisors of n, what is the value of ┬(┬(┬(12)))?
-----------------------------
Answers and thorough explanations are greatly appreciated.
(especially the 2nd one. I have no clue about the 2nd one...)
Thanks!
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1.) Numbers between -11 and 11 that are multiplies of 2:
-10, -8, -6, -4, -2, 0, 2, 4, 6, 8, 10 (11 numbers)
Numbers between -11 and 11 that are multiples of 3:
-9, -6, -3, 0, 3, 6, 9 (7 numbers)
Total amount of numbers = 18 numbers. However, there are 3 numbers common to both (-6, 0, 6). So there are a total of 18 - 3 = 15 nifty numbers between -11 and 11.
#2.)
Remember that divisors and factors are the same thing. The number 12 has factors 1,2,3,4,6,12 or 6 factors. Therefore ┬(12) = 6. Now we are down to ┬(┬(6)) = ┬(4) = 3
-10, -8, -6, -4, -2, 0, 2, 4, 6, 8, 10 (11 numbers)
Numbers between -11 and 11 that are multiples of 3:
-9, -6, -3, 0, 3, 6, 9 (7 numbers)
Total amount of numbers = 18 numbers. However, there are 3 numbers common to both (-6, 0, 6). So there are a total of 18 - 3 = 15 nifty numbers between -11 and 11.
#2.)
Remember that divisors and factors are the same thing. The number 12 has factors 1,2,3,4,6,12 or 6 factors. Therefore ┬(12) = 6. Now we are down to ┬(┬(6)) = ┬(4) = 3
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1. multiple of 2: 2,4,6,8,10 and its negatives so 10
multiple of 3: 3,6,9 6 is already counted so 4
also 0 is a multiple of both so
10 + 5 = 15 nifty numbers
2. numbers of positive divisors of a number n is n = (a^r)(b^s)(c^t) = (r+1)(s+1)(t+1)
12 = 2^2 * 3 so (2+1)(1+1) = 6 divisors
now do it to 6
6 = 2 * 3 (1+1)(1+1) 2*2 = 4
now do it to 4
4 = 2^2 = (2+1)
answer is 3
multiple of 3: 3,6,9 6 is already counted so 4
also 0 is a multiple of both so
10 + 5 = 15 nifty numbers
2. numbers of positive divisors of a number n is n = (a^r)(b^s)(c^t) = (r+1)(s+1)(t+1)
12 = 2^2 * 3 so (2+1)(1+1) = 6 divisors
now do it to 6
6 = 2 * 3 (1+1)(1+1) 2*2 = 4
now do it to 4
4 = 2^2 = (2+1)
answer is 3