The multiples of 4 are the numbers that can be expressed as 4 times an integer. This sequence starts with 0, 4, 8, 12, 16, and continues indefinitely in both the positive and negative directions, such as -4, -8, -12, etc. Essentially, any number that can be divided by 4 without leaving a remainder is a multiple of 4.
11 = (42 - 4) - (4 / 4) 12 = (4 + 4) + (√4 + √4) 13 = (42 - 4) + (4 / 4) 14 = (4 + 4 + 4 + √4) 15 = (4 * 4) - (4 / 4) 16 = (4 + 4 + 4 + 4) 17 = (42 + √4) - (4 / 4) 18 = (42 + 4) - (4 - √4) 19 = (42 + 4) - (4 / 4) 20 = (4 * 4) + (√4 + √4)
Here is one set of solutions. The answers here are not unique. 1 = (4*4)/(4*4) 2 = 4/4 + 4/4 3 = (4+4+4)/4 4 = (4-4)*4 + 4 5 = (4*4 + 4) / 4 6 = 4 + (4+4)/4 7 = 4 + 4 - 4/4 8 = 4 + 4 + 4 - 4 9 = 4 + 4 + 4/4 10 = (44 - 4)/4
3 simple solutions (there are more): (4-4)/4 + 4 = 4 4*(4-4)+ 4 = 4 4-((4-4)/4)) = 4
I'll start it, but I'm not going to finish it! 4 X 4 / 4 - 4 = 0 44 / 44 = 1 4 / 4 + 4 / 4 = 2 (4 + 4 + 4) / 4 = 3 (4 / 4)4 X 4 = 4 (4 / 4)4 + 4 = 5 (4 + 4) / 4 + 4 = 6 44 / 4 - 4 = 7 4 + 4 + 4 - 4 = 8 4 / 4 + 4 + 4 = 9 (44 - 4) / 4 = 10 (44 + 4) / 4 = 12 4! - 44 / 4 = 13 4! / 4 + 4 + 4 = 14 44 / 4 + 4 = 15 4 + 4 + 4 + 4 = 16 4 X 4 + 4 / 4 = 17 4 / √4 + 4 * 4 = 18 4! - 4 + 4 - 4 = 20 4! / 4 + 4 X 4 = 22 4 X 4 + 4 + 4 = 24 44 - 4 * 4 = 28 (4 + 4 / 4)! / 4 = 30 44 / (4 + 4) = 32 4! + 44 / 4 = 35 4! X 4! / 4 / 4 = 36 44 - 4 / √4 = 42 44 - 4 / 4 = 43 44(4/4) = 44 4 / 4 + 44 = 45 44 + 4 / √4 = 46 (4 + 4 + 4) X 4 = 48 √4 * 4 + 44 = 52 4 X 4 X 4 - 4 = 60 44 / 4 - √4 = 62 4(4 - 4/4) = 64 44 / 4 + √4 = 66 4 X 4 X 4 + 4 = 68 4! X 4! / (4 + 4) = 72 (4 X 4 + 4) X 4 = 80 (4 - 4 / 4)4 = 81 √4 * 44 - 4 = 84 √4 * 44 - √4 = 86 44 + 44 = 88 √4 * 44 + √4 = 90 √4 * 44 + 4 = 92 4! X 4 / 4 X 4 = 96
4/4 + 4/4 = 2 (one) 4 - (4+4)/4 = 2 (two) (4*4)/(4+4) = 2 (three) 4*(4/(4+4)) = 2 (four) ((4+4)/4) mod 4 = 2 (five) 4 + √4 - √4 - √4 = 2 (six) ((√4)/(√4)) + ((√4)/(√4)) = 2 (seven) 4 + 4 - 4 - √4 = 2 (eight) (√4)^4 / (4+4) = 2 (nine) (√(4*4)) / (4+4) = 2 (ten)
You have to get a calculator and keep adding 8
1 2 3 4 4+1 4+2 4+3 4+4 4+4+1 4+4+2 4+4+3 4+4+4 4+4+4+1 4+4+4+2 4+4+4+3 4+4+4+4 4+4+4+4+1 4+4+4+4+2 4+4+4+4+3 4+4+4+4+4 4+4+4+4+4+1 4+4+4+4+4+2 4+4+4+4+4+3 4+4+4+4+4+4 4+4+4+4+4+4+1 4+4+4+4+4+4+2 4+4+4+4+4+4+3 4+4+4+4+4+4+4 4+4+4+4+4+4+4+1 4+4+4+4+4+4+4+2 4+4+4+4+4+4+4+3 4+4+4+4+4+4+4+4 4+4+4+4+4+4+4+4+1 4+4+4+4+4+4+4+4+2 4+4+4+4+4+4+4+4+3 4+4+4+4+4+4+4+4+4 4+4+4+4+4+4+4+4+4+1 4+4+4+4+4+4+4+4+4+2 4+4+4+4+4+4+4+4+4+3 4+4+4+4+4+4+4+4+4+4 4+4+4+4+4+4+4+4+4+4+1 4+4+4+4+4+4+4+4+4+4+2 4+4+4+4+4+4+4+4+4+4+3 4+4+4+4+4+4+4+4+4+4+4 4+4+4+4+4+4+4+4+4+4+4+1 4+4+4+4+4+4+4+4+4+4+4+2 4+4+4+4+4+4+4+4+4+4+4+3 4+4+4+4+4+4+4+4+4+4+4+4 4+4+4+4+4+4+4+4+4+4+4+4+1 4+4+4+4+4+4+4+4+4+4+4+4+2 I hope this is the answer you search for! (because it took some time!)
26
1 = 4*4/(4*4) 2 = 4/4+4/4 3 = (4+4+4)/4 4 = (4-4)/4+4 5 = 4^(4-4)+4 6 = (4+4)/4+4 7 = 4+4-4/4 8 = 4+4+4-4 9 = 4/4+4+4 10 = (4*4+4!)/4 11 = (4+4!)/4+4 12 = (4-4/4)*4 13 = (4+4!+4!)/4 14 = 4!/4+4+4 15 = 4*4-4/4 16 = 4*4+4-4 17 = 4*4+4/4 18 = (4*4!-4!)/4 19 = 4!-(4+4/4) 20 = (4/4+4)*4 21 = 4!+4/4-4 22 = 4!-(4+4)/4 23 = 4!-4^(4-4) 24 = 4*4+4+4 25 = 4!+(4/4)^4 26 = 4!+4!/4-4 27 = 4!+4-4/4 28 = (4+4)*4-4 29 = 4/4+4!+4 30 = (4*4!+4!)/4 31 = (4+4!)/4+4! 32 = 4^4/(4+4) 33 = (4-.4)/.4+4! 34 = 4!/4+4+4! 35 = (4.4/.4)+4! 36 = (4+4)*4+4 37 = 4/.4+4+4! 38 = 44-4!/4 39 = (4*4-.4)/.4 40 = (4^4/4)-4! 41 = (4*4+.4)/.4 42 = 4!+4!-4!/4 43 = 44-4/4 44 = 4*4+4+4! 45 = (4!/4)!/(4*4) 46 = (4!-4)/.4 - 4 47 = 4!+4!-4/4 48 = (4*4-4)*4 49 = 4!+4!+4/4 50 = (4*4+4)/.4 51 = 4!/.4-4/.4 52 = 44+4+4 53 = 44+4/.4 54 = (4!/4)^4/4! 55 = (4!-.4)/.4-4 56 = 4!+4!+4+4 57 = 4/.4+4!+4! 58 = (4^4-4!)/4 59 = 4!/.4-4/4 60 = 4*4*4-4 61 = 4!/.4+4/4 62 = (4!+.4+.4)/.4 63 = (4^4-4)/4 64 = 4^(4-4/4) 65 = 4^4+4/4 66 = (4+4!)/.4-4 67 = (4+4!)/.4+4 68 = 4*4*4+4 69 = (4+4!-.4)/.4 70 = (4^4+4!)/4 71 = (4!+4.4)/.4 72 = (4-4/4)*4! 73 = (.4√4+.4)/.4 74 = (4+4!)/.4+4 75 = (4!/4+4!)/.4 76 = (4!-4)*4-4 77 = (4!-.4)/.4+4! 78 = (4!*.4+4!)/.4 79 = (.4√4-.4)/.4 80 = (4*4+4)*4 81 = (4/4-4)^4 82 = 4!/.4+4!+4 83 = (4!-.4)/.4+4! 84 = (4!-4)*4+4 85 = (4/.4+4!)/.4 86 = (4-.4)*4!-.4 87 = 4!*4-4/.4 88 = 4^4/4+4! 89 90 = (4!/4)!/(4+4) 91 92 = (4!-4/4)*4 93 94 = (4+4!)/.4 + 4! 95 = 4!*4-4/4 96 = 4!*4+4-4 97 = 4!*4+4/4 98 = (4!+.4)*4+.4 99 = (4!+4!-4)/.4 100 = 4*4/(.4*.4)
(4+4+4+4+4+4+4+4+4+4+4+4+4+4+4+4+4+4+4+4+4+4+4+4+4+4+4) ÷ 4 = 27
4 +4+4+4+4+4+4+4+4=40
Sure, using the number 4 four times, you can create the numbers 1 to 20 as follows: 1 = 4 / 4 + 4 - 4 2 = 4 / 4 + 4 / 4 3 = 4 - 4 / 4 + 4 4 = 4 + 4 - 4 - 4 5 = 4 + 4 / 4 6 = 4 + 4 - 4 / 4 7 = 4 + 4 / 4 + 4 8 = 4 + 4 + 4 / 4 9 = (4 + 4) / (4 / 4) 10 = 4 + 4 + 4 - 4 11 = 4 + 4 + 4 / 4 12 = 4 + 4 + 4 + 4 13 = (4 + 4) / 4 + 4 14 = 4 * 4 - 4 / 4 15 = 4 + 4 + 4 + 4 - 4 16 = 4 * 4 - 4 + 4 17 = 4 * 4 + 4 / 4 18 = (4 + 4) * (4 - 4) 19 = 4 * 4 + 4 - 4 20 = 4 * 4 + 4 / 4
The primes required are: 2 = (4+4)/4 3 = (4+4+4)/4 5 = (4+4+4+4+4)/4 7 = (4*4+4+4+4)/4 11 = (4(4*4-4)-4)/4 13 = (4*(4*4-4)+4)/4 17 = (4*4*4+4)/4 19 = (4*(4*4+4)-4)/4 23 = (4*(4*4+4+4)-4)/4 29 = (4*(4*4+4+4+4)+4)/4 31 = (4*4*(4+4)-4)/4 37 = (4*4*(4+4)+4*4+4)/4 41 = (4^4-4*(4*4)+4)/4 43 = (4^4-4(4*4+4+4))/4 47 = (4^4-4*4*4-4)/4 The remainder are left as an exercise. It should be noted that most of these are impossible to express with only six fours without either defining new operators or allowing for facetious, unmathematical cheats such as allowing 44 to be used.
1 = 44 / 442 = 4 * 4 / (4 + 4)3 = (4 + 4 + 4) / 44 = 4 + (4 * (4 - 4))5 = (4 + (4 * 4)) / 46 = 4 + ((4 + 4) / 4)7 = (44 / 4) - 48 = 4 + 4 + 4 - 49 = 4 + 4 + (4 / 4)10 = (44 - 4) / 411 = (4 / 4) + (4 / .4)12 = (4 + 44) / 413 = 4 + ((4 - .4) / .4)14 = (4 * (4 - .4)) - .415 = 4 + (44 / 4)16 = (44 - 4) * .417 = (4 * 4) + (4 / 4)18 = (44 * .4) + .419 = (4 + 4 - .4) / .420 = 4 * (4 + (4 / 4))
(4+4+4+4+4+4+4+4+4+4+4+4+4+4+4+4+4) / 4 = 17
4 4 4 44 4 4 4 44 4 4 4 44 4 4 4 4 4 4 4 4 34 4 4 4
0 = 44-44 1 = 44/44 or (4+4)/(4+4) or (4/4) / (4/4) 2 = 4/4+4/4 3 = (4+4+4)/4 4 = 4*(4-4)+4 5 = (4*4+4)/4 6 = 4*.4+4.4 7 = 44/4-4 8 = 4+4.4-.4 9 = 4/4+4+4 10 = 44/4.4 11 = 4/.4+4/4 12 = (44+4)/4 13 = 4-44/4 14 = 4*(4-.4)-.4 15 = 44/4+4 16 = .4*(44-4) 17 = 4/4+4*4 18 = 44*.4+.4 19 = 4!-4-4/4 20 = 4*(4/4+4) 21 = (4.4+4)/.4 22 = 44*sqrt(4)/4 23 = (4*4!-4)/4 24 = 4*4+4+4 25 = (4*4!+4)/4 26 = 4/.4+4*4 27 = 4-4/4+4! 28 = 44-4*4 29 = 4/.4/.4+4 30 = (4+4+4)/.4 31 = (4!+4)/4+4! 32 = 4*4+4*4 33 = (4+4! 34 = 44-4/.4 -6736 = 44-4-4 37 = (sqrt(4)+4!)/holy(4)+4! 38 = 44-4!/4 799740 = 44-sqrt(4*4) 42 = sqrt(4)+44-4 43 = 44-4/4 797945 = 4/4+44 46 = 44-sqrt(4)+4 47 = 4!+4!-4/4 48 = 4*(4+4+4) 49 = (4!-4.4)/.4 50 = 4!/4+44 52= (4!*4)-44 58= (sqrt(4)*(4!+4))+sqrt(4) 72= (4!*4)-4!*4o (small zero uptop four to the zero power = 1)