Form 3512: Fill & Download for Free

With or without repeating digits?With repeating digits:All multiples of 4 end in an even digit. (even digit in the units)If the unit digit is 0, 4, or 8, the next to last digit (the digit in the tens) must be even. If the unit digit is 2 or 6, the tens digit shall be odd.So, any of these five digits can be in the tens, and for each digit in the tens, there is only one possible digit in the units. The five possible ending combinations are 12, 24, 32, 44, 52.Then, whatever goes in the hundreds and tousands is independent, and you have five digits to choose. So the answer is [math]5\times5\times5=\boxed{125}[/math]A list of all of them:1112 1124 1132 1144 1152 1212 1224 1232 1244 1252 1312 1324 1332 1344 1352 1412 1424 1432 1444 1452 1512 1524 1532 1544 1552 2112 2124 2132 2144 2152 2212 2224 2232 2244 2252 2312 2324 2332 2344 2352 2412 2424 2432 2444 2452 2512 2524 2532 2544 2552 3112 3124 3132 3144 3152 3212 3224 3232 3244 3252 3312 3324 3332 3344 3352 3412 3424 3432 3444 3452 3512 3524 3532 3544 3552 4112 4124 4132 4144 4152 4212 4224 4232 4244 4252 4312 4324 4332 4344 4352 4412 4424 4432 4444 4452 4512 4524 4532 4544 4552 5112 5124 5132 5144 5152 5212 5224 5232 5244 5252 5312 5324 5332 5344 5352 5412 5424 5432 5444 5452 5512 5524 5532 5544 5552 Whit no repeating numbers:Still the two last digits must show it is divisible by 4. But we must discard 44. So we have four different endings.Whatever ending we choose, we have used two digits of the five available. So we have only 3 possible choices for the hundreds, and 2 possible choices for the thousands.So the answer is [math]4\times3\times2=\boxed{24}[/math]A list of all of them:1324 1352 1432 1452 1524 1532 3124 3152 3412 3452 3512 3524 4132 4152 4312 4352 4512 4532 5124 5132 5312 5324 5412 5432 

The discovery of exoplanets has kicked into high gear in the last decade or so. At last count, we have 4,057 confirmed discoveries of exoplanets. Kepler went out of commission last year but the pace of discovery hasn’t slowed down, as it gets replaced by other satellites & space telescopes like NASA’s Transiting Exoplanet Survey Satellite (TESS), which is going to continue the mission in the transitory period before James Webb Space Telescope takes Kepler’s spot in 2021. European Characterizing Exoplanets Satellite (CHEOPS) is all set to launch later in the year as well, with the mandate to spot exoplanets.Recently, I reviewed new research, which states that there could be different types of exoplanets out there where life can thrive even more than here on Earth. New discoveries in deep space are not only enlightening us on the creation & evolution of our universe, in some cases, it is also challenging the fundamental laws of Science that we base our studies on.Case in point, the recent discovery of the exoplanet GJ 3512b — a gas giant with a little under half the mass of Jupiter. The surprising fact, however, is that it revolves around a red dwarf star GJ 3512, which is only 250x the mass of its planet. The mass ratio of this red dwarf, 30 light-years from Earth, negates the current theories about the comparable mass of a star to its planets.The current theory suggests that small fragments of dust and gas around the newborn stars slowly collect more material forming planets that slowly grow larger in size over time. Basically this means that a large-sized star would host planets proportionate to its size, while smaller star systems would have smaller worlds.TRAPPIST-1, an ultracool red dwarf star lying only 40 light-years away from Earth is a good example, with a mass of only 8% of our Sun and therefore hosts only seven Earth-sized exoplanets. Now compare the ratio of the biggest gas giant in our Solar system, Jupiter to Sun, where the latter is 16 times the diameter of former and more than 1000 times as massive. It was a well-established hypothesis, till now, that the size of planets in any planetary system was determined by the size of their Star.The research team suggests that the GJ 3512b might have formed as a result of a different mechanism called Gravitational collapse — which in the past has been pitted as the possible explanation of how gas giants get started. It states that a pocket of gas becomes so dense that the gravity of the planet causes it to collapse into a planet. But it raises other questions like:“Why hasn’t the planet continued to grow and migrate closer to the star in this case?” ~ Christoph Mordasini, an author of the studyThe astronomers will continue to decode the mystery as we wait for a more viable answer. The gas giant GJ 3512b was discovered using an infrared spectrograph installed at the Calar Alto Observatory in Spain and the research paper was published in the journal Science.

As you are not allowed to use each digit more than once, and the 3 is fixed (the number 4000 doesn’t intervene as it has zeros) then you have repeated variations without repetition of the numbers [math]\{1, 2, 4, 5, 6\}[/math] taken three at a time or:[math]RV_{5,3} = {5!\over2!} = {5\cdot4\cdot3} = [/math]60Indeed, the numbers list is short that can easily be enumerated in full:3124, 3125, 3126, 3142, 3145, 3146, 3152, 3154, 3156, 3162, 3164, 3165,3214, 3215, 3216, 3241, 3245, 3246, 3251, 3254, 3256, 3261, 3264, 32653412, 3415, 3416, 3421, 3425, 3426, 3451, 3452, 3456, 3461, 3462, 34653512, 3514, 3516, 3521, 3524, 3526, 3541, 3542, 3546, 3561, 3562, 35643612, 3614, 3615, 3621, 3624, 3625, 3641, 3642, 3645, 3651, 3652, 3654.