Tuesday, August 7th, 2012
Problem 49:
The arithmetic sequence, 1487, 4817, 8147, in which each of the terms increases by 3330, is unusual in two ways: (i) each of the three terms are prime, and, (ii) each of the 4-digit numbers are permutations of one another.
There are no arithmetic sequences made up of three 1-, 2-, or 3-digit primes, exhibiting this property, but there is one other 4-digit increasing sequence.
What 12-digit number do you form by concatenating the three terms in this sequence?
Tuesday, August 7th, 2012
Problem 49: Find the last ten digits of the series, 1^1 + 2^2 + 3^3 + … + 1000^1000.
Tuesday, August 7th, 2012
Problem 47: Find the first four consecutive integers to have four distinct primes factors. What is the first of these numbers?
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?
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.
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?
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:
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?
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?
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 d1000000