A conversation in school
Andrei decides to do some random computation in his mind. He starts out with the number x . First, he multiplies his number by 3 . Then, he squares his new number. Lastly, he subtracts 8 times the square of his original number from the new number. His final result is the units digit of 3 2 0 1 6 + 2 ( 2 5 1 0 0 0 ) . When Kostya tries to figure out the original number, he discovers that there are multiple answers. Andrei then tells Kostya that his starting number was his date of birth (Kostya did not know it). What number did Andrei start with?
This is adapted from a problem in the IMO.
The answer is 1.
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2 solutions
How did you conclude that The Last digit of 3 2 0 1 6 is 1? How did you conclude that by saying that 2016 is divisible by 4? Please explain. Thanks! :)
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3 1 m o d 1 0 = 3
3 2 m o d 1 0 = 9
3 3 m o d 1 0 = 7
3 4 m o d 1 0 = 1
3 5 m o d 1 0 = 3
Since the pattern repeats after 5 , the unit's digit of any power y of 3 is given by :- y ≡ n ( m o d 4 ) where for n = 0 , 1 , 2 , 3 , the unit's digit is 1 , 3 , 9 , 7 respectively.
Since 2 0 1 6 ≡ 0 ( m o d 4 ) Thus, the unit's digit of 3 2 0 1 6 is 1
The resultant number = (3x)^2-8x^2 = x^2 Unit digit of the given expression = 1+0= 1 X^2= 1 X= +1 or -1 X>0 (given) X=1
First of all, we compute the last digit of 3 2 0 1 6 , as 2 0 1 6 is divisible by 4 then the unit digit of this number is 1 . Now, the last digit of 2 5 1 0 0 0 is 5 and if it is multiplied by 2 then the last digit is 0 . So, the last digit of the expression 3 2 0 1 6 + 2 ( 2 5 1 0 0 0 ) is 1 . Now, according to the question, if the required number be x ( 3 x ) 2 − 8 x 2 = 1 x 2 = 1 x = ± 1 As, the number is a date of birth so the required answer is 1