Number of solutions to x^2+y^2+z^2=81 / x2+y2+z2=81

The number of integer solutions to the equation x^2 + y^2 + z^2 = 81 are
(A)72 (B)15 (C)30 (D)102 (E)120

Answer: D

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Partition of Integers

How many ways can we partition the number 504 using 4 non negative Integers?

Answer: 507 C  3
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Numbers -Word Problems

The number 210 can be written as the sum of consecutive positive integers in several ways. When written as the sum of the greatest possible number of consecutive positive integers, what is the largest of these integers?

Sum to n terms 1,2,3,.........n is n(n+1) / 2.This will give us greated number of consecutive positive integers.

n(n+1)/2 = 210
so n(n+1)= 420
20*21= 420
so answer is 20

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Euclid's Algorithm

A pair of positive integers (x, y) satisfies the equation 31x + 29 y = 1125. What is x + y? ans - x+y=37

31x+29y=1125

31x+29y=1 we get for x=-14 and y=15

general solution is x =-14*1125+29*t y = 15*1125-31t

14*1125/29 <= t<= 15*1125/31 t is an integer
we get t as 544

plug in t back in the equations above
to get x =26 and y=11
so x+y=37

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