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**1+2=3**(generalized) Math Club 3/19/2012 Powerpoint Templates**Definition**Weshould all know that 1+2=3. A politenumberis a numberthatcanbewritten as a sum of positive consecutivenumbers. For example, 4+5+6+7 is 22, so 22 is a politenumber. Through a few challenges, wewill know everythingthereis to know about politenumbers by the end of thispowerpoint.**Challenge 1**Show that 3 is a politenumber.**Challenge 2**Show that 9 is a politenumber.**Challenge 3**Is 4 a politenumber?**Challenge 4**Find all politenumbers.**Challenge 5**Show that all oddnumbers are polite.**Challenge 6**Show thatanynumber divisible by 3 ispolite.**Challenge 7**Provethatanynumber divisible by 7 ispolite.**Challenge 8**Let besomeoddnumber. Provethat for any, infinitelymany multiples of are polite. (in otherwords, above a certain point, all multiples of are polite)**Challenge 9**Challenge 8 istooweak. Let besomeoddnumber. Provethat for any, ALL multiples of are politenumbers.**Challenge 10**All numberswith an odd factor are polite. Whatnumbers are left?**Final Challenge (11)**Show that a power of twocannotbepolite.