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Divisibility rule
A divisibility rule is a shorthand way of determining whether a given number is divisible by a ... as 2^{n} or 5^{n}, in which n is a positive integer, just ex... 

If N is a positive integer, what is the last digit of 1! + : GMAT ...
May 12, 2010 2:50 am. Dartastic wrote: If N is a positive integer, what is the lastdigit of 1! + 2! + ... +N!? 1) N is divisible by 4 2) (N^2 + 1)/5 is an ... 

Patterns in the Last Digits of the Positive Powers of Two  Exploring ...
The last digit — the ones place — of a decimal integer d is the ... You can also relate the positive powers of two to their ending digits based on ... Ends in 4: \ mbox{\footnotesize{\displaystyle{n \equiv 2 \pmod{ ... 2^{40+100k}, 40, 776 ... . for (i= 1,10,print(i,": ",4*5^(i1))) 1: 4 2: 20 3: 100 4: 500 5: 2500 ... 
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Problem 2  Art of Problem Solving
feet to either side of the first and last trees in the row before the ends of the yard. ... What is Joshua's correct answer (the units digit of $2008^{2008}$ )? ... . be the set of positive integers that are the product of two consecutive integers. ... . is an integer, find the number of possible values of $n$ . ... . Problem 40 ... Pr... 

Square Number  from Wolfram MathWorld
Actually, the basis set for representing positive integers with positive squares is { 0,1,4,9,16,25,36,64 , so 49 need never be used. ... n, square partitions. 10, 4. 50, 104. 100, 1116. 150, 6521. 200, 27482 ... To find the possible last digits for a square number, write n=10a+b for the number ... .. 99066^2=9814072356. (40) ... 

Archived Problems  Project Euler
Work out the first ten digits of the sum of the following onehundred 50digit numbers. ... The following iterative sequence is defined for the set of positiveintegers: ... . However, when n = 40, 40^{2} + 40 + 41 = 40(40 + 1) + 41 is divisible by 41, and certainly when n = 41, 41² ... .. Find the last ten digits of the series, 1^{1} + 2^{2} + 3... 

What's Special About This Number?  Stetson University
18 is the only positive number that is twice the sum of its digits. 19 is the ... 50 is the smallest number that can be written as the sum of of 2 squares in 2 ways. ... 103 has the property that placing the last digit first gives 1 more than triple it. ... 105 is the largest number n known with the property that n  2^{k} is prime for k>... 

Integers and Sequence (Solution)  COINHEIST  MIT
12 42 18 40 30 24 20; 2 1 132 42 429 14; 7 9 1 8 5 3 10 4; 92 117 70 145 35 1 22 ... (the number of positive integers less than 2013 that are divisible by 100, but not ... same and the last two digits the same), 89 (A005100 and A074317) + 50 ( we ... integers n for which the sum of the n smallest positive integers evenly divides ... .. 

Types of Numbers  Math Goodies
The positive integers, 1, 2, 3... , are called the natural numbers or counting numbers. ... Irrational numbers are expressible only as decimal fractions where the digits ... An abundant number is a number n for which the sum of divisors σ(n) >2n, or, ... .. 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, etc., are all composite numbers... 