The number of elements in the set { (a, b) : | vedclass

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(C) Given the equation $2a^2 + 3b^2 = 35$ for integers $a, b \in Z$.We test possible values for $b^2$ such that $3b^2 < 35$:If $b^2 = 0$,$2a^2 = 35$ (

Vedclass · Accepted Answer

$8$

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(C) Given the equation $2a^2 + 3b^2 = 35$ for integers $a, b \in Z$.We test possible values for $b^2$ such that $3b^2 If $b^2 = 0$,$2a^2 = 35$ (no integer solution for $a$).If $b^2 = 1$,$2a^2 = 35 - 3 = 32 \implies a^2 = 16 \implies a = \pm 4$. This gives pairs $(4, 1), (4, -1), (-4, 1), (-4, -1)$.If $b^2 = 4$,$2a^2 = 35 - 12 = 23$ (no integer solution for $a$).If $b^2 = 9$,$2a^2 = 35 - 27 = 8 \implies a^2 = 4 \implies a = \pm 2$. This gives pairs $(2, 3), (2, -3), (-2, 3), (-2, -3)$.If $b^2 = 16$,$3b^2 = 48 > 35$ (no further solutions).Thus,the set contains $8$ elements: $(4, 1), (4, -1), (-4, 1), (-4, -1), (2, 3), (2, -3), (-2, 3), (-2, -3)$.

The number of elements in the set $\{ (a, b) : 2a^2 + 3b^2 = 35, a, b \in Z \} $,where $Z$ is the set of all integers,is

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