I just saw this question in one exam. Please help me solve it. I am not able to find any clue on where to begin.
2025-01-13 00:06:53.1736726813
How to find the value of this expression?
135 Views Asked by Gaurang Tandon https://math.techqa.club/user/gaurang-tandon/detail At
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Using the following identity:
$$a^4 + 4\cdot 3^4 = (a^2 + 2 \cdot 3^2 - 2\cdot 3\cdot a)(a^2 + 2 \cdot 3^2 + 2\cdot 3\cdot a) = (a(a-6) + 18)(a(a+6)+18)$$
Most of the terms cancel out and you are left with:
$$\frac{58(64)+18}{4(-2)+18} = \frac{3730}{10} = 373$$
As KprimeX mentioned, this flows from the Sophie Germain Identity.
As it seems to be a multiple choice question, we can find the answer without much computing: all factors have the form $$(2n)^4+324=4(4n^4+81).$$ Since there are as many factors in the numerator as in the denominator, the $4$s cancel out, which results in a fraction with odd numerator and denominator, hence this fraction must simplify to an odd number. There's only one odd number in the proposed answers.