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4) If the pattern below were continued, what number would be directly below 100?
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The numbers are an arithmetic sequence.
- Each row contains 2 more numbers than the previous row (1, 3, 5, 7 ...)
- The rows contain sequential numbers that are reversed in order from the previous row.
- You must find out which row 100 appears in and in what position in the row.
- Notice that the highest number in each row (1,4,9,16) are the squares of the row numbers (1,2,3,4).
- This means that 100 appears in the
N = √100 = 10th row.
- Also, because 100 is a square of 10 itself, it is the highest number in row 10.
- Every even numbered row is reversed so the 10th row looks like this:
    100 99 98 97 ...
- So the next row below it is not reversed and looks like this:
       100   99   98   97 ...
101 102 103 104 ...
- So the next number under the 100 is 102
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5) How many numbers between 100 and 1,000 have exactly 3 factors?
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- All numbers have 2 factors: 1 and the number itself.
Prime numbers have these factors only.
- The only numbers that have exactly 1 more factor are squares. For example the factors of 25 are 1, 5 and 25. But only numbers whose square roots are primes, otherwise there will be additional factors.
- Numbers that have prime square roots:
11x11 = 121
13x13 = 169
17x17 = 289
19x19 = 361
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23x23 = 529
29x29 = 841
31x31 = 961
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- 7 numbers between 100 and 1000 that have 3 factors.
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