The factorial notation!
Number Theory
Level
2
Find the sum of the last two digits of the expression: 0!+1!+2!+.....+9!+10!
Do not use a calculator.
0
5
10
Too tedious
13
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1 solution
10!=10(9)(8).....(2)(1)=.......00
0!=1 , 1!=1 ,2!=2 ,3!=6 ,4!=24 ,5!=120 ,6!=720
For 7!,since it is equal to 6!(7),we only need to consider the last two digits of 6!,then multiply it by 7 to get: 7!=......40(last 2 digits)
Likewise, for 8!, it is equal to ....40(8)=......20
For 9!, it is equal to ......20(9)=......80
Thus, the sum of the last 2 digits of the expression is obtainable from the sum of the last 2 digits of each term in the expression: 01+01+02+06+24+20+20+40+20+80+00=34+180=214, and the sum of the last two digits of the original expression is:1+4=5(Q.E.D)