Discussion
If 21168 = \(2^{a} \times 3^{b} \times 7^{c}\), where a, b and c are natural numbers, then what is the value of (4a − 5b + c) ?
Answer: D
The value of (4a − 5b + c) is 3.
Factorise 21168 by repeated division:
\[ \begin{aligned} 21168 &= 7 \times 3024 \\ &= 7^{2} \times 432 \\ &= 7^{2} \times 16 \times 27 \\ &= 2^{4} \times 3^{3} \times 7^{2} \end{aligned} \]
So a = 4, b = 3 and c = 2.
\[ 4a - 5b + c = 16 - 15 + 2 = 3 \]
Check: \(16 \times 27 \times 49 = 21168\). Miscounting a power changes the result sharply; for example a = 3 would give 12 − 15 + 2 = −1, which is not even an option.
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