all solutions will appear shortly
1. a natural number n can be replaced by ab if a + b = n
is it possible, if we start with n=22 to get to 2001??
prove.
solution:
Yes, it can.
note that n = (n-1)+1, so from n we can get (n-1).
now we only need to get any number larger than 2001, and descend it to 2001
one by one.
example: 22 = 11+11 ® 121 = 60+61 ® 3660 ® 3659 ®...® 2001
2.What is the unit digit of the sum 1!+2!+3!+...+15!?
solution:
lets write a few down:
1!=1, 2!=2, 3!=6, 4!=24, 5!=120, 6!=720, ...
since 5! ends in a zero all the following will also end in a zero.
therefore the unit digit, is the unit digit of 1+2+3+6+4=13 or 3
3.palindromes, read the same forward and backwards.
Find the sum of all four-digit positive integer palindromes.
solution:
listing all the palindromes will look something like this-
[1001,1111,1221,...,9999]
if we add the first and the last palindromes we get 11000, adding the second and the one
before last we also get 11000.
since there are 10 palindromes starting with each digit from 1 to 9,
there are 90 palindromes. therefore the sum is (45)(11000)=495000
4.Find the greatest n for which 12^n evenly divides by 20!.
solution:
will be added shortly
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