1. The factors of 28 (not including itself) are 1, 2, 4, 7
and 14. Astonishingly these factors add up to give 28. Because 28 has this special property it is called a Perfect Number. 28 = 1^{3} + 3^{3} and so is also the sum of the first two
odd cubes. Calculate 1^{3} + 3^{3} + 5^{3} + 7^{3}. Find all the factors of this number. Is
the number perfect?
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2. Since 3^{2} + 4^{2} = 5^{2} does
it follow that 3^{3} + 4^{3} + 5^{3} = 6^{3} ? Check it out !!
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3. Continue with these calculations and .... be amazed!
1x9 + 2 = 11
12x9 + 3 = 111
123x9 + 4 = 1111
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4.
3 x 37 = 111 and 1+1+1 = 3
6 x 37 = 222 and 2+2+2 = 6
9 x 37 = 333 and 3+3+3 = 9
...
continue this pattern and be amazed!
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5. 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: 99+91=8 so it doesn't divide by 11.
Let's check 2013: 31+02=0 so it does divide by 11 ( since 0 divides
exactly by 11).
Let's check: 723456789: 98+76+54+32+7=11 so it does divide by 11.
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6. In the decade
19901999 three of the years were prime:
1993 , 1997 and this year 1999.
Since all primes other than 2 and 5 end in
either 1, 3, 7 or 9 then this decade has been RICH
in primes.
The most prime years possible in a decade is
four. The last such VERY RICH decade was 18001899:
1801, 1803, 1807 and 1809 are all prime.
An amazing fact is that there has been no VERY RICH decade for one hundred
years.
In fact the current
RICH decade is the first since 17801789:
1783, 1787 and 1789 are all prime.
So how long do you have to wait until the next RICH or
VERY RICH decade? Will it be in your lifetime?
The good news is that if you are of school
age just now you may just experience the next VERY RICH decade.
However the bad news is that you will definitely
not experience another RICH decade like the present one.
So when are they? Search for yourself in the
The Prime List.
