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I think an easier way to solve this is to look closely at that 90%. we know that 90%, or 90 individuals, are academic professors or tenured or both, so the other 10 individuals must be professional professors non tenured. remembering that there are 40 professional professors, then 30 of those 40 are tenured, and there you have your answer
I think neither 1 nor 2 are enough idependently or together cuase here me have 3 unknown and we have 2 equations with those 3 uknown. Then, the interval information given does not add any value. A I wrong?
Total combinations of triangles = 15C3 = 455 But to get equidistant dots = 15/3 = 5 different set of dots that form equilateral triangle. Now subtract to get 455-5 = 450
Yeah too much steps, could find the prime factor of 15, which are 3 and 5. Now 759325 = 3373 *25 and 5^2= 25. Thus, K=2. Substitute into the equation and you'll get 1, which is the answer.
I'm impressed by your level of speaking fluency and clarity, congratulations!!! May I ask something ? . Which is the most practical way to detect if a question will take too long to be answered? Are there any hacks or shortcuts to know this? Thanks!
After seeing all of your GREAT explanations (thx 4 that), I only have one question, and I'm afraid is kind of a dumb one (lol). Last problem, the one with the equilateral triangles... The 15 factorial (15!) was 15 x 14 x 13. My question is: Doesn't the factorial (!) mean 15 x 14 x 13 x 12 x 11 x 10 and so on til 1? I mean, how do you know that you needed to stop at 13? Thanks
When you take a combination of xCy, you’re dividing X!/Y!(X-Y)! In this case 15!/3!(12!), you can express 15! As 15x14x13x12… and 12! As 12x11x10… when you divide 15!/12! Everything after 13 cancels out, because it is also part of 12!.
The divisibility rules of all numbers from 201 to 300 201: The number is a multiple of 3 and 67 at the same time 202: The number is a multiple of 2 and 101 at the same time 203: The number is a multiple of 7 and 29 at the same time 204: The number is a multiple of 3, 4 and 17 at the same time 205: The number is a multiple of 5 and 41 at the same time 206: The number is a multiple of 2 and 103 at the same time 207: The number is a multiple of 9 and 23 at the same time 208: The number is a multiple of 13 and 16 at the same time 209: The number is a multiple of 11 and 19 at the same time 210: The number is a multiple of 3, 7 and 10 at the same time 211: The difference between 21 times the last digit and the rest of the number is a multiple of 211 212: The number is a multiple of 4 and 53 at the same time 213: The number is a multiple of 3 and 71 at the same time 214: The number is a multiple of 2 and 107 at the same time 215: The number is a multiple of 5 and 43 at the same time 216: The number is a multiple of 8 and 27 at the same time 217: The number is a multiple of 7 and 31 at the same time 218: The number is a multiple of 2 and 109 at the same time 219: The number is a multiple of 3 and 73 at the same time 220: The number is a multiple of 11 and 20 at the same time 221: The number is a multiple of 13 and 17 at the same time 222: The number is a multiple of 2, 3 and 37 at the same time 223: The sum of 27 times the last digit and the rest of the number is a multiple of 223 224: The number is a multiple of 7 and 32 at the same time 225: The number is a multiple of 9 and 25 at the same time 226: The number is a multiple of 2 and 113 at the same time 227: The difference between 68 times the last digit and the rest of the number is a multiple of 227 228: The number is a multiple of 3, 4 and 19 at the same time 229: The sum of 23 times the last digit and the rest of the number is a multiple of 229 230: The number is a multiple of 10 and 23 at the same time 231: The number is a multiple of 3, 7 and 11 at the same time 232: The number is a multiple of 8 and 29 at the same time 233: The sum of 70 times the last digit and the rest of the number is a multiple of 233 234: The number is a multiple of 2, 9 and 13 at the same time 235: The number is a multiple of 5 and 47 at the same time 236: The number is a multiple of 4 and 59 at the same time 237: The number is a multiple of 3 and 79 at the same time 238: The number is a multiple of 2, 7 and 17 at the same time 239: The sum of 24 times the last digit and the rest of the number is a multiple of 239 240: The number is a multiple of 3 and 80 at the same time 241: The difference between 24 times the last digit and the rest of the number is a multiple of 241 242: The number is a multiple of 2 and 121 at the same time 243: The sum of 73 times the last digit and the rest of the number is a multiple of 243 244: The number is a multiple of 4 and 61 at the same time 245: The number is a multiple of 5 and 49 at the same time 246: The number is a multiple of 2, 3 and 41 at the same time 247: The number is a multiple of 13 and 19 at the same time 248: The number is a multiple of 8 and 31 at the same time 249: The number is a multiple of 3 and 83 at the same time 250: The number ends in 000, 250, 500 or 750 251: The difference between 25 times the last digit and the rest of the number is a multiple of 251 252: The number is a multiple of 4, 7 and 9 at the same time 253: The number is a multiple of 11 and 23 at the same time 254: The number is a multiple of 2 and 127 at the same time 255: The number is a multiple of 3, 5 and 17 at the same time 256: The last 8 digits are a multiple of 256; the 10,000,000s digit is even and the last 7 digits are a multiple of 256 or the 10,000,000s digit is odd and the last 7 digits are 128 times an odd number 257: The difference between 77 times the last digit and the rest of the number is 0 or a multiple of 257 258: The number is a multiple of 2, 3 and 43 at the same time 259: The number is a multiple of 7 and 37 at the same time 260: The number is a multiple of 13 and 20 at the same time 261: The number is a multiple of 9 and 29 at the same time 262: The number is a multiple of 2 and 131 at the same time 263: The sum of 79 times the last digit and the rest of the number is a multiple of 263 264: The number is a multiple of 3, 8 and 11 at the same time 265: The number is a multiple of 5 and 53 at the same time 266: The number is a multiple of 2, 7 and 19 at the same time 267: The number is a multiple of 3 and 89 at the same time 268: The number is a multiple of 4 and 67 at the same time 269: The sum of 27 times the last digit and the rest of the number is a multiple of 269 270: The number is a multiple of 10 and 27 at the same time 271: The difference between 27 times the last digit and the rest of the number is a multiple of 271 272: The number is a multiple of 16 and 17 at the same time 273: The number is a multiple of 3, 7 and 13 at the same time 274: The number is a multiple of 2 and 137 at the same time 275: The number is a multiple of 11 and 25 at the same time 276: The number is a multiple of 3, 4 and 23 at the same time 277: The difference between 83 times the last digit and the rest of the number is a multiple of 277 278: The number is a multiple of 2 and 139 at the same time 279: The number is a multiple of 9 and 31 at the same time 280: The number is a multiple of 7 and 40 at the same time 281: The difference between 28 times the last digit and the rest of the number is a multiple of 281 282: The number is a multiple of 2, 3 and 47 at the same time 283: The sum of 85 times the last digit and the rest of the number is a multiple of 283 284: The number is a multiple of 4 and 71 at the same time 285: The number is a multiple of 3, 5 and 19 at the same time 286: The number is a multiple of 2, 11 and 13 at the same time 287: The number is a multiple of 7 and 41 at the same time 288: The number is a multiple of 9 and 32 at the same time 289: The sum of 29 times the last digit and the rest of the number is a multiple of 289 290: The number is a multiple of 10 and 29 at the same time 291: The number is a multiple of 3 and 97 at the same time 292: The number is a multiple of 4 and 73 at the same time 293: The sum of 88 times the last digit and the rest of the number is a multiple of 293 294: The number is a multiple of 2, 3 and 49 at the same time 295: The number is a multiple of 5 and 59 at the same time 296: The number is a multiple of 8 and 37 at the same time 297: The number is a multiple of 11 and 27 at the same time 298: The number is a multiple of 2 and 149 at the same time 299: The number is a multiple of 13 and 23 at the same time 300: The number is a multiple of 3 and 100 at the same time
The divisibility rules of every number from 101 to 200: 101: The difference between 10 times the last digit and the rest of the number is a multiple of 101 102: The number is a multiple of 2, 3 and 17 at the same time 103: The sum of 31 times the last digit and the rest of the number is a multiple of 103 104: The number is a multiple of 8 and 13 at the same time 105: The number is a multiple of 3, 5 and 7 at the same time 106: The number is a multiple of 2 and 53 at the same time 107: The difference between 32 times the last digit and the rest of the number is a multiple of 107 108: The number is a multiple of 4 and 27 at the same time 109: The sum of 11 times the last digit and the rest of the number is a multiple of 109 110: The number is a multiple of 10 and 11 at the same time 111: The number is a multiple of 3 and 37 at the same time 112: The number is a multiple of 7 and 16 at the same time 113: The sum of 34 times the last digit and the rest of the number is a multiple of 113 114: The number is a multiple of 2, 3 and 19 at the same time 115: The number is a multiple of 5 and 23 at the same time 116: The number is a multiple of 4 and 29 at the same time 117: The number is a multiple of 9 and 13 at the same time 118: The number is a multiple of 2 and 59 at the same time 119: The number is a multiple of 7 and 17 at the same time 120: The number is a multiple of 3 and 40 at the same time 121: The difference between 12 times the last digit and the rest of the number is a multiple of 121 122: The number is a multiple of 2 and 61 at the same time 123: The number is a multiple of 3 and 41 at the same time 124: The number is a multiple of 4 and 31 at the same time 125: The number ends in 000, 125, 250, 375, 500, 625, 750 or 875 126: The number is a multiple of 2, 7 and 9 at the same time 127: The difference between 38 times the last digit and the rest of the number is a multiple of 127 128: The last 7 digits are a multiple of 128; the 1,000,000s digit is even and the last 6 digits are a multiple of 128 or the 1,000,000s digit is odd and the last 6 digits are 64 times an odd number 129: The number is a multiple of 3 and 43 at the same time 130: The number is a multiple of 10 and 13 at the same time 131: The difference between 13 times the last digit and the rest of the number is a multiple of 131 132: The number is a multiple of 3, 4 and 11 at the same time 133: The number is a multiple of 7 and 19 at the same time 134: The number is a multiple of 2 and 67 at the same time 135: The number is a multiple of 5 and 27 at the same time 136: The number is a multiple of 8 and 17 at the same time 137: The difference between 41 times the last digit and the rest of the number is a multiple of 137 138: The number is a multiple of 2, 3 and 23 at the same time 139: The sum of 14 times the last digit and the rest of the number is a multiple of 139 140: The number is a multiple of 7 and 20 at the same time 141: The number is a multiple of 3 and 47 at the same time 142: The number is a multiple of 2 and 71 at the same time 143: The number is a multiple of 11 and 13 at the same time 144: The number is a multiple of 9 and 16 at the same time 145: The number is a multiple of 5 and 29 at the same time 146: The number is a multiple of 2 and 73 at the same time 147: The number is a multiple of 3 and 49 at the same time 148: The number is a multiple of 4 and 37 at the same time 149: The sum of 15 times the last digit and the rest of the number is a multiple of 149 150: The number is a multiple of 3 and 50 at the same time 151: The difference between 15 times the last digit and the rest of the number is a multiple of 151 152: The number is a multiple of 8 and 19 at the same time 153: The number is a multiple of 9 and 17 at the same time 154: The number is a multiple of 2, 7 and 11 at the same time 155: The number is a multiple of 5 and 31 at the same time 156: The number is a multiple of 3, 4 and 13 at the same time 157: The difference between 47 times the last digit and the rest of the number is a multiple of 157 158: The number is a multiple of 2 and 79 at the same time 159: The number is a multiple of 3 and 53 at the same time 160: The last 5 digits are a multiple of 160; the 10,000s digit is even and the last 4 digits are a multiple of 160 or the 10,000s digit is odd and the last 4 digits are 80 times an odd number 161: The number is a multiple of 7 and 23 at the same time 162: The number is a multiple of 2 and 81 at the same time 163: The sum of 49 times the last digit and the rest of the number is a multiple of 163 164: The number is a multiple of 4 and 41 at the same time 165: The number is a multiple of 3, 5 and 11 at the same time 166: The number is a multiple of 2 and 83 at the same time 167: The difference between 50 times the last digit and the rest of the number is a multiple of 167 168: The number is a multiple of 3, 7 and 8 at the same time 169: The sum of 17 times the last digit and the rest of the number is a multiple of 169 170: The number is a multiple of 10 and 17 at the same time 171: The number is a multiple of 9 and 19 at the same time 172: The number is a multiple of 4 and 43 at the same time 173: The sum of 52 times the last digit and the rest of the number is a multiple of 173 174: The number is a multiple of 2, 3 and 29 at the same time 175: The number is a multiple of 7 and 25 at the same time 176: The number is a multiple of 11 and 16 at the same time 177: The number is a multiple of 3 and 59 at the same time 178: The number is a multiple of 2 and 89 at the same time 179: The sum of 18 times the last digit and the rest of the number is a multiple of 179 180: The number is a multiple of 9 and 20 at the same time 181: The difference between 18 times the last digit and the rest of the number is a multiple of 181 182: The number is a multiple of 2, 7 and 13 at the same time 183: The number is a multiple of 3 and 61 at the same time 184: The number is a multiple of 8 and 23 at the same time 185: The number is a multiple of 5 and 37 at the same time 186: The number is a multiple of 2, 3 and 31 at the same time 187: The number is a multiple of 11 and 17 at the same time 188: The number is a multiple of 4 and 47 at the same time 189: The number is a multiple of 7 and 27 at the same time 190: The number is a multiple of 10 and 19 at the same time 191: The difference between 19 times the last digit and the rest of the number is a multiple of 191 192: The number is a multiple of 3 and 64 at the same time 193: The sum of 58 times the last digit and the rest of the number is a multiple of 193 194: The number is a multiple of 2 and 97 at the same time 195: The number is a multiple of 3, 5 and 13 at the same time 196: The number is a multiple of 4 and 49 at the same time 197: The difference between 59 times the last digit and the rest of the number is a multiple of 197 198: The number is a multiple of 2, 9 and 11 at the same time 199: The sum of 20 times the last digit and the rest of the number is a multiple of 199 200: The number ends in 000, 200, 400, 600 or 800
