问题1.找到以下数字的多维数据集：
(i)7(ii)12
(iii)16(iv)21
(v)40(vi)55
(vii)100(viii)302
(ix)301
解决方案：
(i) 7
Cube of 7 is
7 = 7× 7 × 7 = 343
(ii) 12
Cube of 12 is
12 = 12× 12× 12 = 1728
(iii) 16
Cube of 16 is
16 = 16× 16× 16 = 4096
(iv) 21
Cube of 21 is
21 = 21 × 21 × 21 = 9261
(v) 40
Cube of 40 is
40 = 40× 40× 40 = 64000
(vi) 55
Cube of 55 is
55 = 55× 55× 55 = 166375
(vii) 100
Cube of 100 is
100 = 100× 100× 100 = 1000000
(viii) 302
Cube of 302 is
302 = 302× 302× 302 = 27543608
(ix) 301
Cube of 301 is
301 = 301× 301× 301 = 27270901
问题2.编写所有1到10之间的自然数的多维数据集，并验证以下语句：
(i)所有奇数自然数的立方体都是奇数。
(ii)所有偶数自然的立方体都是偶数。
解决方案：
Finding the Cube of natural numbers up to 10
13 = 1 × 1 × 1 = 1
23 = 2 × 2 × 2 = 8
33 = 3 × 3 × 3 = 27
43 = 4 × 4 × 4 = 64
53 = 5 × 5 × 5 = 125
63 = 6 × 6 × 6 = 216
73 = 7 × 7 × 7 = 343
83 = 8 × 8 × 8 = 512
93 = 9 × 9 × 9 = 729
103 = 10 × 10 × 10 = 1000
Hence, we conclude
(i) Cubes of all odd natural numbers are odd.
(ii) Cubes of all even natural numbers are even.
问题3.遵循以下模式：
1 3 = 1
1 3 + 2 3 =(1 + 2) 2
1 3 + 2 3 + 3 3 =(1 + 2 + 3) 2
写下接下来的三行，并按照上述模式计算1 3 + 2 3 + 3 3 +…+ 9 3的值。
解决方案：
From the given pattern,
13 + 23 + 33 +…+ 93
13 + 23 + 33 +…+ n3 = (1+2+3+…+n) 2
So when n = 10
13 + 23 + 33 +…+ 93 + 103 = (1+2+3+…+10) 2
= (55)2 = 55×55 = 3025
问题4.写出5个自然数的多维数据集，它们是3的倍数，并验证以下内容：
“自然数的立方是3的倍数是27的倍数”
解决方案：
3, 6, 9, 12, and 15 are the first 5 natural numbers which are multiple of 3
So, let’s see how to find the cube of 3, 6, 9, 12 and 15
33 = 3 × 3 × 3 = 27
63 = 6 × 6 × 6 = 216
93 = 9 × 9 × 9 = 729
123 = 12 × 12 × 12 = 1728
153 = 15 × 15 × 15 = 3375
As we can see that all the cubes are divisible by 27
Hence, “The cube of a natural number which is a multiple of 3 is a multiple of 27’
问题5.编写5个自然数的多维数据集，其形式为3n +1(例如4，7，10，…)，并验证以下内容：
“自然数形式为3n + 1的立方体是相同形式的自然数，即当除以3时，余数为1′
解决方案：
4, 7, 10, 13, and 16 are the first 5 natural numbers in the form of (3n + 1)
So now, let us find the cube of 4, 7, 10, 13 and 16
43 = 4 × 4 × 4 = 64
73 = 7 × 7 × 7 = 343
103 = 10 × 10 × 10 = 1000
133 = 13 × 13 × 13 = 2197
163 = 16 × 16 × 16 = 4096
When all the above cubes are divided by ‘3’ leaves the remainder of 1.
Hence, the statement “The cube of a natural number of the form 3n+1 is a natural number of the same form i.e. when divided by 3 it leaves the remainder 1’ is true.
问题6.写出3n + 2形式的5个自然数(即5、8、11…。)，并验证以下内容：
“自然数形式为3n + 2的立方是相同形式的自然数，即当它被3除时，余数为2′
解决方案：
5, 8, 11, 14 and 17 are the first 5 natural numbers in the form (3n + 2)
So now, let us find the cube of 5, 8, 11, 14 and 17 is
53 = 5 × 5 × 5 = 125
83 = 8 × 8 × 8 = 512
113 = 11 × 11 × 11 = 1331
143 = 14 × 14 × 14 = 2744
173 = 17 × 17 × 17 = 4913
When all the above cubes are divided by ‘3’ leaves the remainder of 2
Hence, the statement “The cube of a natural number of the form 3n+2 is a natural number of the same form i.e. when it is dividend by 3 the remainder is 2’ is true.
问题7.编写5个自然数的多维数据集，其中7个整数为7，然后验证以下内容：
“ 7的倍数的立方是7 3的倍数。
解决方案：
7, 14, 21, 28 and 35 are the first 5 natural numbers which are multiple of 7
So now, let us find the cube of 7, 14, 21, 28 and 35
73 = 7 × 7 × 7 = 343
143 = 14 × 14 × 14 = 2744
213 = 21× 21× 21 = 9261
283 = 28 × 28 × 28 = 21952
353 = 35 × 35 × 35 = 42875
We can see that all the above cubes are multiples of 73(343) as well.
Hence, the statement“The cube of a multiple of 7 is a multiple of 73 is true.
问题8.下列哪个是完美的立方体?
(i)64(ii)216
(iii)243(iv)1000
(v)1728(vi)3087
(vii)4608(viii)106480
(ix)166375(x)456533
解决方案：
(i) 64
Finding the factors of 64
64 = 2 × 2 × 2 × 2 × 2 × 2 = 26 = (22)3 = 43
Hence, it’s a perfect cube.
(ii) 216
Finding the factors of 216
216 = 2 × 2 × 2 × 3 × 3 × 3 = 23 × 33 = 63
Hence, it’s a perfect cube.
(iii) 243
Finding the factors of 243
243 = 3 × 3 × 3 × 3 × 3 = 35 = 33 × 32
Hence, it’s not a perfect cube.
(iv) 1000
Finding the factors of 1000
1000 = 2 × 2 × 2 × 5 × 5 × 5 = 23 × 53 = 103
Hence, it’s a perfect cube.
(v) 1728
Finding the factors of 1728
1728 = 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3 = 26 × 33 = (4 × 3 )3 = 123
Hence, it’s a perfect cube.
(vi) 3087
Finding the factors of 3087
3087 = 3 × 3 × 7 × 7 × 7 = 32 × 73
Hence, it’s not a perfect cube.
(vii) 4608
Finding the factors of 4608
4608 = 2 × 2 × 3 × 113
Hence, it’s not a perfect cube.
(viii) 106480
Finding the factors of 106480
106480 = 2 × 2 × 2 × 2 × 5 × 11 × 11 × 11
Hence, it’s not a perfect cube.
(ix) 166375
Finding the factors of 166375
166375= 5 × 5 × 5 × 11 × 11 × 11 = 53 × 113 = 553
Hence, it’s a perfect cube.
(x) 456533
Finding the factors of 456533
456533= 11 × 11 × 11 × 7 × 7 × 7 = 113 × 73 = 773
Hence, it’s a perfect cube.
问题9.以下哪几是偶数个自然数的立方?
216、512、729、1000、3375、13824
解决方案：
(i) 216 = 23 × 33 = 63
It’s a cube of even natural number.
(ii) 512 = 29 = (23)3 = 83
It’s a cube of even natural number.
(iii) 729 = 33 × 33 = 93
It’s not a cube of even natural number.
(iv) 1000 = 103
It’s a cube of even natural number.
(v) 3375 = 33 × 53 = 153
It’s not a cube of even natural number.
(vi) 13824 = 29 × 33 = (23)3 × 33 = 83×33 = 243
It’s a cube of even natural number.
问题10.以下哪些是奇数自然数的多维数据集?
125、343、1728、4096、32768、6859
解决方案：
(i) 125 = 5 × 5 × 5 × 5 = 53
It’s a cube of odd natural number.
(ii) 343 = 7 × 7 × 7 = 73
It’s a cube of odd natural number.
(iii) 1728 = 26 × 33 = 43 × 33 = 123
As 12 is even number. It’s not a cube of odd natural number.
(iv) 4096 = 212 = (26)2 = 642
As 64 is an even number. Its not a cube of odd natural number.
(v) 32768 = 215 = (25)3 = 323
As 32 is an even number. It’s not a cube of odd natural number.
(vi) 6859 = 19 × 19 × 19 = 193
It’s a cube of odd natural number.
问题11.乘以以下数字以使乘积成为理想的立方的最小数是多少?
(i)675(ii)1323
(iii)2560(iv)7803
(v)107811(vi)35721
解决方案：
(i) 675
Finding the factors of 675.
675 = 3 × 3 × 3 × 5 × 5
= 33 × 52
Hence, we need to multiply the product by 5.
(ii) 1323
Finding the factors of 1323
1323 = 3 × 3 × 3 × 7 × 7
= 33 × 72
Hence, we need to multiply the product by 7 to make it a perfect cube.
(iii) 2560
Finding the factors of 2560
2560 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 5
= 23 × 23 × 23 × 5
Hence, we need to multiply the product by 5 × 5 = 25 to make it a perfect cube.
(iv) 7803
Finding the factors of 7803
7803 = 3 × 3 × 3 × 17 × 17
= 33 × 172
Hence, we need to multiply the product by 17 to make it a perfect cube.
(v) 107811
First find the factors of 107811
107811 = 3 × 3 × 3 × 3 × 11 × 11 × 11
= 33 × 3 × 113
Hence, we need to multiply the product by 3 × 3 = 9 to make it a perfect cube.
(vi) 35721
First find the factors of 35721
35721 = 3 × 3 × 3 × 3 × 3 × 3 × 7 × 7
= 33 × 33 × 72
Hence, we need to multiply the product by 7 to make it a perfect cube.