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@@ -37,21 +37,77 @@ PROBLEMS:
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7. By listing the first six prime numbers: 2, 3, 5, 7, 11, and 13, we can see that the 6th prime is 13.
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What is the Nth prime number?
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8. Find the consecutive k digits in a number N that have the largest product.
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9. A Pythagorean triplet is a set of three natural numbers, a < b < c, for which,
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a^2 + b^2 = c^2
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There exists exactly one Pythagorean triplet for which a + b + c = 1000.
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Find the product abc.
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10. Find sum of all prime numbers below 2 million.
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11. In the given 20x20 grid, find 4 adjacent numbers (horizontally, vertically or diagonally) that have the largest product.
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12. The sequence of triangle numbers is generated by adding the natural numbers. So the 7th triangle number would be 1 + 2 + 3 + 4 + 5 + 6 + 7 = 28. The first ten terms would be:
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1, 3, 6, 10, 15, 21, 28, 36, 45, 55, ...
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Let us list the factors of the first seven triangle numbers:
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1: 1
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3: 1,3
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6: 1,2,3,6
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10: 1,2,5,10
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15: 1,3,5,15
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21: 1,3,7,21
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28: 1,2,4,7,14,28
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We can see that 28 is the first triangle number to have over five divisors.
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What is the value of the first triangle number to have over five hundred divisors?
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13. Work out the first 10 digits of the sum of the given hundred 50 digit numbers.
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14. The following iterative sequence is defined for the set of positive integers:
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n → n/2 (n is even)
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n → 3n + 1 (n is odd)
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Using the rule above and starting with 13, we generate the following sequence:
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13 → 40 → 20 → 10 → 5 → 16 → 8 → 4 → 2 → 1
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Which starting number, under one million, produces the longest chain?
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15. Starting from top left corner of a 20x20 grid how many routes are there to reach the bottom left corner?
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16. 2^15 = 32768 and the sum of its digits is 3 + 2 + 7 + 6 + 8 = 26.
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What is the sum of the digits of the number 2^1000?
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17. If the numbers 1 through 1000 were written in words, how many total letters would be used?
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18. By starting at the top of the triangle below and moving to adjacent numbers on the row below, the maximum total from top to bottom is 23.
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7 4
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2 4 6
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8 5 9 3
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That is, 3 + 7 + 4 + 9 = 23.
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Find the maximum total from top to bottom of the triangle below:
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75
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95 64
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17 47 82
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18 35 87 10
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20 04 82 47 65
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19 01 23 75 03 34
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88 02 77 73 07 63 67
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99 65 04 28 06 16 70 92
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41 41 26 56 83 40 80 70 33
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41 48 72 33 47 32 37 16 94 29
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53 71 44 65 25 43 91 52 97 51 14
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70 11 33 28 77 73 17 78 39 68 17 57
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91 71 52 38 17 14 91 43 58 50 27 29 48
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63 66 04 68 89 53 67 30 73 16 69 87 40 31
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04 62 98 27 23 09 70 98 73 93 38 53 60 04 23
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20. n! means n × (n − 1) × ... × 3 × 2 × 1
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For example, 10! = 10 × 9 × ... × 3 × 2 × 1 = 3628800,
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and the sum of the digits in the number 10! is 3 + 6 + 2 + 8 + 8 + 0 + 0 = 27.
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