<p>1.) If a number 12k can be formed into a cube of a positive integer, that is 12k = m^3 , what could be the least possible value of k ?</p>
<p>Answer: 18
Since 12k = 2 x 2 x 3 x k = ( )^3 , the least value of k must be 2 x 3 x 3 = 18</p>
<p>My Question: I don't understand the book explanation. Can someone explain?</p>
<p>This is a more difficult problem than most.</p>
<p>Let N be the number. We have N = 12<em>k, and N = n</em>n*n.</p>
<p>Factor the number 12 – the “best” factorization is that as the product of primes. You have 12 = 2<em>2</em>3. The factorization of a number into powers of primes is a unique factorization.</p>
<p>For N to be a cube we need at least another factor of 2, and two factors of 3. Then we would have N = (2<em>2</em>2)<em>(3</em>3<em>3). Or N =(2</em>3)^^3 = 6^^3. So k = 2<em>3</em>3</p>
<p>To try out your understanding consider a variant of the problem. N = 14<em>k and N is again a cube. Here 14 = 2</em>7. That’s the unique factorization of 14 into primes. So k would need to be (minimally) (2<em>2)</em>(7*7).</p>
<p>A simpler variant is this. Suppose N=6<em>k, and instead of N being a cube it’s a square. So first we write 6 = 2</em>3. And now k needs to be a product of 2<em>3; this so N=(2</em>2)<em>(3</em>3), And for this example k = 2*3 = 6.</p>
<p>haha, hi again.</p>
<p>12k = 2 x 2 x 3 x k. you want to have a m^3 from a 12k, so at least you should have a (2^3)x(3^3) from the 12k, since 12 is already a multiple of 2 and 3, right? that is why k is what makes up (2^3)x(3^3) together with 12, which is 2 x 3 x 3 = 18</p>
<p>I hope it helps :)</p>
<p>Here are some examples of perfect cubes to help clarify:</p>
<p>2<em>2</em>2=8
3<em>3</em>3=27
4<em>4</em>4=64
5<em>5</em>5=125</p>
<p>2<em>2</em>2<em>3</em>3<em>3=8</em>27=216
2<em>2</em>2<em>5</em>5*5=1000</p>
<p>Now here are some integers that are not perfect cubes:</p>
<p>2<em>2=4 is not a perfect cube, but if we multiply by 2 we get a perfect cube:
2</em>2*2=8</p>
<p>2<em>2</em>2<em>3</em>3=72 is not a perfect cube, but multiplying by 3 makes it a perfect cube:
2<em>2</em>2<em>3</em>3*3=216</p>
<p>Ok. One more example, I’m going to pick a random integer, say 84, and find the least integer I need to multiply by to get a perfect cube. We start by factoring 84:</p>
<p>84=2<em>42=2</em>2<em>21=2</em>2<em>3</em>7.</p>
<p>I see that I need to multiply by 2<em>3</em>3<em>7</em>7 = 882 to get a perfect cube. </p>
<p>We would then get the perfect cube 84*882=74,088.</p>
<p>You can check this in your calculator by taking the cube root. Note that 74,088=42^3</p>