Problem 47: Find the first four consecutive integers to have four distinct primes factors. What is the first of these numbers?
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Project Euler – Problem 47
Tuesday, August 7th, 2012
Project Euler – Problem 46
Tuesday, August 7th, 2012
Problem 46: What is the smallest odd composite that cannot be written as the sum of a prime and twice a square?
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Project Euler – Problem 45
Tuesday, August 7th, 2012
Problem 45:
Triangle, pentagonal, and hexagonal numbers are generated by the following formulae:
Triangle Tn=n(n+1)/2 1, 3, 6, 10, 15, ... Pentagonal Pn=n(3n1)/2 1, 5, 12, 22, 35, ... Hexagonal Hn=n(2n1) 1, 6, 15, 28, 45, ...
It can be verified that T285 = P165 = H143 = 40755.
Find the next triangle number that is also pentagonal and hexagonal.
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Project Euler – Problem 44
Tuesday, August 7th, 2012
Problem 44: Pentagonal numbers are generated by the formula, Pn=n(3*n-1)/2. The first ten pentagonal numbers are:
1, 5, 12, 22, 35, 51, 70, 92, 117, 145, ...
It can be seen that P4 + P7 = 22 + 70 = 92 = P8. However, their difference, 70 – 22 = 48, is not pentagonal.
Find the pair of pentagonal numbers, Pj and Pk, for which their sum and difference is pentagonal and D = |Pk – Pj| is minimised; what is the value of D?
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Project Euler – Problem 43
Monday, August 6th, 2012
Problem 43: The number, 1406357289, is a 0 to 9 pandigital number because it is made up of each of the digits 0 to 9 in some order, but it also has a rather interesting sub-string divisibility property.
Let d1 be the 1st digit, d2 be the 2nd digit, and so on. In this way, we note the following:
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Project Euler – Problem 42
Monday, August 6th, 2012
Problem 42: The nth term of the sequence of triangle numbers is given by, tn = ½n(n+1);
Using this word list, convert each letter in a word to a number corresponding to its alphabetical position and adding these values we form a word value. For example, the word value for SKY is 19 + 11 + 25 = 55 = t10. If the word value is a triangle number then we shall call the word a triangle word.
How many are triangle words?
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Project Euler – Problem 41
Monday, August 6th, 2012
Problem 41: We shall say that an n-digit number is pandigital if it makes use of all the digits 1 to n exactly once. For example, 2143 is a 4-digit pandigital and is also prime.
What is the largest n-digit pandigital prime that exists?
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Project Euler – Problem 40
Monday, August 6th, 2012
Problem 40: An irrational decimal fraction is created by concatenating the positive integers:
0.123456789101112131415161718192021...It can be seen that the 12th digit of the fractional part is 1.
If dn represents the nth digit of the fractional part, find the value of the following expression.
d1 x d10 x d100 x d1000 x d10000 x d100000 x d1000000Continued reading >
Project Euler – Problem 39
Sunday, August 5th, 2012
Problem 39: If p is the perimeter of a right angle triangle with integral length sides, {a,b,c}, there are exactly three solutions for p = 120.
{20,48,52}, {24,45,51}, {30,40,50}
For which value of p 1000, is the number of solutions maximised?
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Project Euler – Problem 38
Sunday, August 5th, 2012
Problem 38: Given the following example:
192 x 1 = 192 192 x 2 = 384 192 x 3 = 576
By concatenating each product we get the 1 to 9 pandigital, 192384576. We will call 192384576 the concatenated product of 192 and (1,2,3)
What is the largest 1 to 9 pandigital 9-digit number that can be formed as the concatenated product of an integer with (1,2, … , n) where n > 1?
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