First slide

Combinations

Question

Number of points having position vector $a \hat{i} + b \hat{j} + c \hat{k}$ , where a, b, c ∈ {1, 2, 3, 4, 5} such that 2^a + 3^b + 5^c is divisible by 4 is

Moderate

A

140
B

70
C

100
D

None of these

Solution

$\begin{array}{l} 4 m = 2^{a} + 3^{b} + 5^{c} = 2^{a} + (4 - 1)^{b} + (4 + 1)^{c} \\ 4 m = 4 k + 2^{a} + (- 1)^{b} + (1)^{c} \end{array}$
∴ a = 1, b = even, c = any number
a ≠ 1, b = odd, c = any number
∴ Required number = 1 × 2 × 5 × 4 × 3 × 5 = 70

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