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Find the smallest integer a such that 945a is a

Berikut ini adalah pertanyaan dari wooooodshshshsh pada mata pelajaran Matematika untuk jenjang Sekolah Menengah Pertama

Find the smallest integer a such that 945a is a perfect cube.

Jawaban dan Penjelasan

Berikut ini adalah pilihan jawaban terbaik dari pertanyaan diatas.

There is no smallest integer asuch that945a is a perfect cube.
However, if a or the perfect cube is a positive integer, the smallest integer a such that 945ais a perfect cube is1225.

Explanation

Let bbe the root of the perfect cube which is the value of945a. Thus,

\large\text{$\begin{aligned}b^3=945a\end{aligned}$}

By prime factorization, we get:

\large\text{$\begin{aligned}b^3&=3\cdot315\cdot a\\&=3\cdot3\cdot105\cdot a\\&=3\cdot3\cdot3\cdot35\cdot a\\&=3^3\cdot5\cdot7\cdot a\\\end{aligned}$}

The integer ashould be in the form of{}\pm 35^{(3k-1)}, k=1,2,3,{\dots}. Unlike perfect squares, perfect cubes could be negative. Hence, temporary solution we can get is:

\large\text{$\begin{aligned}&a=-\left(35^{(3k-1)}\right )\\&\quad k=1,2,3,{\dots}\end{aligned}$}

The larger the k is, the smaller the a will be.

THEREFORE, there is no smallest integer asuch that945a is a perfect cube.

However, if aor the perfect cube is apositive integer, the smallest integer asuch that945a is a perfect cube is:

\large\text{$\begin{aligned}a&=35^{(3\cdot1\:-\:1)}\\&=35^2\\&=\bf1225\end{aligned}$}

\blacksquare

 

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Last Update: Mon, 22 Aug 22
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