Modular Arithmetic
The digits of a positive integer are four consecutive integers in decreasing order when read from left to right.
What is the sum of the possible remainders when is divided by 37?
The answer is 217.
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1 solution
Preferred Short Method : The number is of the form (n+3)(n+2)(n+1)(n).......................{Four Consecutive Integers} Therefore the number= 1000(n+3)+100(n+2)+10(n+1)+n = 1111*n+3210.......................{On Simplification} But the condition on n is 0<=n<=6..................(The reason being the smallest number can be 3210 and the largest number can be 9876)
On Further Noticing we come to know that, 1111=1(mod 37) This implies 1111 n= 1 n (mod 37).............(1) And 3210= 28 (mod 37).........................(2)
Adding (1) and (2) we get, 1111*n+3210= 28+n (mod 37)
Substituting n=0,1,2,3,4,5,6 we get, The possible remainders are 28+0,28+1,28+2,........,28+6 = 28+29+30+31+32+33+34 = 217.
Long Hit and Trail Method: Divide the following numbers by 37 and add the remainders: 3210 (Remainder=28) 4321 (29) 5432 (30) 6543(31) 7654 (32) 8765 (33) 9876. (34).