Just for fun!-XVI

Number Theory Level 3

Find the unit digit of the sum ( 1 ! ) 2 + ( 2 ! ) 2 + ( 3 ! ) 2 + . . . + ( 2007 ! ) 2 { (1!) }^{ 2 }+{ (2!) }^{ 2 }+{ (3!) }^{ 2 }+...+{ (2007!) }^{ 2 }


The answer is 7.

Swapnil Das by Swapnil Das , 20, India

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2 solutions

Tay Yong Qiang
Aug 20, 2015

5 ! = 120 5!=120 and thus factorials beyond 5 ! 5! are also mutiples of 10 10 , which have 0 0 as their unit digit when squared. The unit digit of the required sum is thus ( ( 1 ! ) 2 + ( 2 ! ) 2 + ( 3 ! ) 2 + ( 4 ! ) 2 ) m o d 10 = 7 m o d 10 ((1!)^2+(2!)^2+(3!)^2+(4!)^2)mod 10=\boxed{7}mod 10 .

3 Helpful 0 Interesting 0 Brilliant 0 Confused
Paola Ramírez
Aug 21, 2015

After 5 ! 5! all the numbers end in zero so the unit digit is 1 + 4 + 6 + 576 = 58 7 1+4+6+576=58\boxed{7}

2 Helpful 0 Interesting 0 Brilliant 0 Confused

In fact the sum is: 1 + 4 + 36 + 576 = 617 1+4+36+576=617 . But we got it!!! ;-) \Large \text{;-)}

Cleres Cupertino - 5 years, 9 months ago

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