How can you tell if a number divides exactly by 2? Answer: If it ends in an even number. How can you tell if a number divides exactly by 5? Answer: If it ends in a 0 or a 5. These facts are very familiar however... How can you tell if a number divides by 11? Answer: Alternately add and subtract the digits from right to left. If the answer divides by 11 then so does the original number Let's check 1999: 9-9+9-1=8 so it doesn't divide by 11. Let's check 2013: 3-1+0-2=0 so it does divide by 11 ( since 0 divides exactly by 11). Let's check: 723456789: 9-8+7-6+5-4+3-2+7=11 so it does divide by 11. Now try some of your own... --- 153 So what's amazing about 153 ?
Not only that...Try the following sum: 1+2+3+4+5+6+7+8+9+10+11+12+13+14+15+16+17
And if that's not enough:Try the following sum: 1! + 2! + 3! + 4! + 5! ( ! is the factorial sign )
This last fact is shared by very few other numbers.Try the following sum: 13 + 53 + 33
They are called the three digit Armstrong Numbers and there are only four of them. In increasing order they are: 153, ***, 371 and 407. Can you find the missing one? --- The primes 3, 5, 7 form an Arithmetic Sequence. There is a constant difference of 2 between one prime and the next. However the next term of the sequence is 9 which is not a prime number. So we have found only three prime numbers in arithmetic sequence. The primes 5, 11, 17, 23, 29 form an Arithmetic Sequence with constant difference 6. The next term 35 is not prime, but five primes forming an arithmetic sequence beats three primes! So what's the world record? If you start with the prime 100996972469714247637786655587969840329509324689190041803603417758904341703348882159067229719 and use a constant difference of 210 you will produce the record breaking arithmetic sequence of TEN primes discovered by Manfred Toplic on 2 March 1998. More information. Can you find an example to beat the five prime sequence? Here's The Prime List. --- SORT -- REVERSE -- SUBTRACT Let's take the year 1999: SORT: 9991 REVERSE: 1999 SUBTRACT: 7992 And now repeat the process with 7992:
Notice that a constant number has been reached.9972 2799 7173
7731 1377 6354
6543 3456 3087
8730 0378 8352
8532 2358 6174
7641 1467 6147
This is Amazing Number Fact 24 so let's process 24:
Notice that this time a cycle has been entered:42 24 18
81 18 63
63 36 27
72 27 45
54 45 09
90 09 81
81 18 63
63 -> 27 -> 45 -> 09 -> 81 -> repeat Are there other constants? Are there other cycles? Investigate! And what has this to do with Kaprekar? --- Sometimes whether you add or whether you multiply makes no difference. Familiar examples are: 2 x 2 = 2 + 2 and 0 x 0 = 0 + 0. However the following may not be so familiar:
What are the next few examples? Check them.11/2 x 3 = 11/2 + 3 = 41/2 11/3 x 4 = 11/3 + 4 = 51/3 11/4 x 5 = 11/4 + 5 = 61/4 Can you generalise using algebra and prove your generalisation? It is also possible for multiplication and subtraction to be swapped:
Check the next few examples.1 x 1/2 = 1 - 1/2 = 1/2 2 x 2/3 = 2 - 2/3 = 11/3 3 x 3/4 = 3 - 3/4 = 21/4 Again try to generalise using algebra and prove your generalisation. Can you discover examples swapping division and addition? --- This continues or does it? 12 = 1 22 = (1+1)2 = 1+2+1 32 = (1+1+1)2 = 1+2+3+2+1 42 = (1+1+1+1)2 = 1+2+3+4+3+2+1 this continues... or does it? 12=1 112=121 1112=12321 11112=1234321 this continues... or does it? --- NINE DIGITS The nine digits are: 1, 2, 3, 4, 5, 6, 7, 8 and 9 (0 is still on holiday) What is special about these multiplications?
Now calculate these:12 x 483 = 5796 27 x 198 = 5346 42 x 138 = 5796 39 x 186 = 7254
Now calculate these and look carefully at the answers and be amazed:18 x 297 48 x 159 28 x 157 4 x 1738 4 x 1963 are there others? 3 x 51249876 9 x 16583742 6 x 32547891 are there others?
To find out more try The Largest Known Prime"ORLANDO, Florida, June 30, 1999 -- Nayan Hajratwala, a participant in the Great Internet Mersenne Prime Search (GIMPS), has discovered the first known million-digit prime number using software written by George Woltman and the distributed computing technology and services of Scott Kurowski's company, Entropia.com, Inc. The prime number: 26,972,593 -1, contains 2098960 digits qualifying for the $50,000 award offered by the Electronic Frontier Foundation (EFF). An article is being submitted to an academic journal for consideration. The new prime number, discovered on June 1st, is one of a special class of prime numbers called Mersenne primes. This is only the 38th known Mersenne prime. There is a well-known formula that generates a "perfect" number from a Mersenne prime. A perfect number is one whose factors add up to the number itself. The smallest perfect number is 6 = 1 + 2 + 3. The newly discovered perfect number is 2^6972592 * (2^6972593-1) This number is 4,197,919 digits long!"
A dangerous link is: Download the prime. (Be warned it's over 2MB!) Even more dangerous: Download the Perfect Number (Over 4MB!!) If you want to join the search for a larger prime and possibly win $100000 try: GIMPS --- Seven Multiplications and a bit more
Can you continue this pattern? Check the answers!1 x 7 + 3 = 10 14 x 7 + 2 = 100 142 x 7 + 6 = 1000 1428 x 7 + 4 = 10000 14285 x 7 + 5 = 100000 142857 x 7 + 1 = 1000000 1428571 x 7 + 3 = 10000000 14285714 x 7 + 2 = 100000000 142857142 x 7 + 6 = 1000000000 1428571428 x 7 + 4 = 10000000000 What has the pattern to do with 1/7 as a decimal? --- 
Does this pattern continue? Is there a similar pattern for root 3 , root 5 ...?
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