Sudal's Garage

Question: Surprisingly there are only three numbers that can be written as the sum of fourth powers of their digits:1634 = 14 + 64 + 34 + 44 8208 = 84 + 24 + 04 + 84 9474 = 94 + 44 + 74 + 44As 1 = 14 is not a sum it is not included.The sum of these numbers is 1634 + 8208 + 9474 = 19316.Find the sum of all the numbers that can be written as the sum of fifth powers of their digits. 문제:놀랍게도 각 자릿수의 ..

Question: Consider all integer combinations of ab for 2 ≤ a ≤ 5 and 2 ≤ b ≤ 5: 22=4, 23=8, 24=16, 25=32 32=9, 33=27, 34=81, 35=243 42=16, 43=64, 44=256, 45=1024 52=25, 53=125, 54=625, 55=3125 If they are then placed in numerical order, with any repeats removed, we get the following sequence of 15 distinct terms:4, 8, 9, 16, 25, 27, 32, 64, 81, 125, 243, 256, 625, 1024, 3125How many distinct term..

Question: Starting with the number 1 and moving to the right in a clockwise direction a 5 by 5 spiral is formed as follows:21 22 23 24 25 20 7 8 9 10 19 6 1 2 11 18 5 4 3 12 17 16 15 14 13It can be verified that the sum of the numbers on the diagonals is 101.What is the sum of the numbers on the diagonals in a 1001 by 1001 spiral formed in the same way? 문제:1부터 시작해서 오른쪽으로 움직인 뒤 시계방향으로 숫자를 적어나가는 방..

Question: A unit fraction contains 1 in the numerator. The decimal representation of the unit fractions with denominators 2 to 10 are given:1/2= 0.51/3= 0.(3)1/4= 0.251/5= 0.21/6= 0.1(6)1/7= 0.(142857)1/8= 0.1251/9= 0.(1)1/10= 0.1Where 0.1(6) means 0.166666..., and has a 1-digit recurring cycle. It can be seen that 1/7 has a 6-digit recurring cycle.Find the value of d < 1000 for which 1/d contai..

Question: Euler discovered the remarkable quadratic formula:n2+n+41It turns out that the formula will produce 40 primes for the consecutive integer values 0≤n≤39. However, when n=40,40^2+40+41=40(40+1)+41 is divisible by 41, and certainly when n=41,41^2+41+41 is clearly divisible by 41. The incredible formula n^2−79n+1601 was discovered, which produces 80 primes for the consecutive values 0≤n≤79..