Sum of 5 squares

Probability Level 4

Brilli the ant is considering the complexities of a 5-dimensional universe. She wants to count the number of integer points that are distance n \sqrt{n} away from the origin. Let TnT_n be the set of ordered 5-tuples of integers (a1,a2,a3,a4,a5)(a_1,a_2,a_3,a_4,a_5) such that a12+a22+a32+a42+a52=n.a_1^2 + a_2^2 + a_3^2 + a_4^2 + a_5^2 = n. However, being an ant, she doesn't have enough toes to record numbers that are above 10. Let DnD_n be the units digit of Tn.\vert T_n \vert.

Determine i=15678Di.\sum_{i = 1}^{5678} D_i.

Details and assumptions

Note: You are not asked to find the last digit of the sum, but the sum of all the last digits.

For a set SS, S | S | denotes the number of elements in SS. You can read up on set notation.


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