What is the lowest possible number that have 9 factors? (Please show me the method)?
83369
I am giving you a method which would minimize number of possibilities.\nFactorize the number of factors( in your example 9) in as many way as possible.( in your example 9 and 3*3)\nNow find out the possible numbers ( one for one factor combination) in following way ( we are going to have 2 numbers)\nThen all factor combinations are sorted so that largest one is in first place and lowest one in last place.\nFor a factor combination x1*x2*x3*...xn*...\nthe number will be \n2^(x1-1) * 3^(x2-1) * 5^(x3-1) * ... * pn(xn-1)...\nwhere pn is nth prime number.\nFor our two combinations,\nfor 9-> 2^(9-1)=256\nfor 3*3-> 2^(3-1) * 3^(3-1) = 36\nLowest of these numbers is our result. \nIn our example it is 36.\nLet us see another example lowest number with 12 factors.\n12 factorizes to 12, 4*3, 6*2, 3*2*2\nfor 12-> 2^(12-1)=2048\nfor 4*3-> 2^(4-1) * 3^(3-1) = 72\nfor 6*2-> 2^(6-1) * 3^(2-1) = 96\nfor 3*2*2-> 2^(3-1) * 3^(2-1) * 5^(2-1)=60.\nSo our answer is 60.\nThis method will be if the number of factor cobination  is too large. An way to get good answer( may not be correct) is to start from prime factorization and continue untill collecting first two terms into one causes better result.\n\nThat is in case of 12\nWe will start from 2*2*3 and get 60.\nThen test if 4*3 is better. If not we will take 60 as answer.( as it is 72 so we will).\nThis case we have correct result but modified version may not produce correct result.
455504
I will explain a method that is the most easy on this planet, and a unique one.\n\nI will illustrate with the help of a simple example.\n\nEg. 4-2.\n\nDraw no. line as follows:\n\n<-6--5--4--3--2--1--0--1--2--3--4--5--6--7-->\n\nNumbers left to zero are negative integers\nand to the right are the positive ones.\n\nLocate 4 on this line. Now move 2 steps behind 4 as the operation to be carried out is subtraction.\n\nThis will give answer as 2. \n\nSimilarly for addition move the steps ahead.
709001
Here are 3.  Asking for the 3 lowest common multiples is alot more interesting...  \n\n198,  396,  1188
748322
The numbers for this particular answer are 9 and 10.\n\n9 + 10*3\n9 + 30\n=39
339712
If 3 is a zero then P(3) = 0.\nP(3) = 27 + 36 - 21 - 10 = 32 \nThus 3 is not a zero.\nSince 3 is not a zero, then x-3 cannot be a factor.\nPossible rational factors are the positive and negative factors of 10\n1,2,5,10\nP(5) = 180\nP(10) = 1320\nThere are no rational factors greater than 3.\nAlso, there is not a zero between 3 and 4.\nC.\nThe zeros are -5, -1, and 2\nP(x) = (x+5)(x+1)(x-2)
177874
It is 2 x 5 x 9 = 90\n1 x 90 = 2 x 45 = 3 x 30 = 5 x 18 = 6 x 15 = 9 x 10.\nThese are 12 factors. We had already three of them.\nSo there are nine more.
713027
Just try out small numbers. 1, 2, 3, 4, 6, 8 all work. Once you start doing anything higher, the second factor becomes smaller again, so the others are 96/8, 96/6, 96/4, 96/3, 96/2, 96/1.\nIn all: 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96.
1099830
27\n\n(+7 -4 +8 -3 +9 -2 +10)
157699
Moocow is only partly correct - the smallest cases known have 17 numbers, but it has not been proven that this is the minimum.\n\nFrom the American Scientist:\nFor order 3 [standard-sized Sudoku puzzles,] the minimum number of givens is unknown. Gordon Royle of the University of Western Australia has collected more than 24,000 examples of uniquely solvable grids with 17 givens, and he has found none with fewer than 17, but a proof is lacking.
460189
the answer is the lowest common multiple of 2,3,4,5,6,7,8,9\n\nwhich is 2.2.2.3.3.5.7=2520
893652
with 1q, 2 ways (0,1 n)\nwith 3d, 1 ways (0 n)\nwith 2d, 3 ways (0,1,2 n)\nwith 1d, 5 ways (0,1,2,3,4 n)\nwith 0d, 7 ways (0,1,2,3,4,5,6 n)\n\nTotal of 18 ways
331342
The factors of 186 can be found as follows:\n\n186 is even, so 2 is a factor\n186/2 = 93\n\nBoth 9 and 3 are divisable by 3, so 3 is a factor\n93/3=31\n\n31 is prime, so it's the final factor.\n\nI'm not sure how you'll get anything to add up to 46 though, maybe 36?
376218
Not sure how many you need, but here are the first few:\n2,3,5,7,11,13,17,19,23,29,31, 37,41,43,47,53,59,61,67, 71,73,79,83,89,97,101
875190
Perform a prime factorization of each number:\n16 = 2 x 2 x 2 x 2\n36 = 2 x 2 x 3 x 3\n\nTo find the lowest common multiple, you need to take each prime factor, enough to cover both numbers.  In other words you need four of the 2s and two of the 3s.\n\nLCM(16, 36) = 2 x 2 x 2 x 2 x 3 x 3\nLCM(16, 36) = 144\n\nThe other people might be thinking greatest common factor, but even that answer is wrong since the GCF(16, 36) is 4, not 2.
481134
Since 3, 7, and 8 don't have any common factors, the least common multiple is found by multiplying them together.\n3*7*8 = 168
1553
9*9*9*9*9*9 = 531441\n\n(After all, this is the humor category)
939962
Take it step by step.\n\n37 will go into 223 how many times?  What is left over?  Bring down the 4.  \n\nHow many times can 37 go into that number.  Repeat the steps