Question

He number of 4 digit numbers which are neither perfect squares nor perfect cubes is _____

Answer 1

Solution :-

Number of 4 digit numbers between from 1000 to 9999 (both are included)

= 9999 - 999 = 9000

So, there are 9000 4 digit numbers from 1000 to 9999 (both included)

Perfect square numbers from 1000 to 9999 are as follows.

1) 32² = 1024
2) 33² = 1089
3) 34² = 1156
4) 35² = 1225
5) 36² = 1296
6) 37² = 1369
7) 38² = 1444
8) 39² = 1521
9) 40² = 1600
10) 41² = 1681 
11) 42² = 1764
12) 43² = 1869
13) 44² = 1936
14) 45² = 2025
15) 46² = 2116
16) 47² = 2209
17) 48² = 2304
18) 49² = 2401
19) 50² = 2500
20) 51² = 2601
21) 52² = 2704
22) 53² = 2809
23) 54² = 2916
24) 55² = 3025
25) 56² = 3136
26) 57² = 3249 
27) 58² = 3364
28) 59² = 3481
29) 60² = 3600
30) 61² = 3721
31) 62² = 3844
32) 63² = 3969
33) 64² = 4096
34) 65² = 4225
35) 66² = 4356
36) 67² = 4489
37) 68² = 4624
38) 69² = 4761
39) 70² = 4900
40) 71² = 5041 
41) 72² = 5184 
42) 73² = 5329
43) 74² = 5476
44) 75² = 5625
45) 76² = 5776
46) 77² = 5929
47) 78² = 6084
48) 79² = 6241
49) 80² = 6400
50) 81² = 6521
51) 82² = 6724
52) 83² = 6889
53) 84² = 7056
54) 85² = 7225
55) 86² = 7396
56) 87² = 7569
57) 88² = 7744
58) 89² = 7921
59) 90² = 8100
60) 91² = 8281
61) 92² = 8464
62) 93² = 8649
63) 94² = 8836
64) 95² = 9025
65) 96² = 9216
66) 97² = 9409
67) 98² = 9604
99) 99² = 9801

These are 68 four digit perfect square numbers from 1000 to 9999

Perfect cube numbers from 1000 to 9999 are as follows.

1) 10³ = 1000
2) 11³ = 1331
3) 12³ = 1728
4) 13³ = 2197
5) 14³ = 2744
6) 15³ = 3375
7) 16³ = 4096
8) 17³ = 4913
9) 18³ = 5832
10) 19³ = 6859
11) 20³ = 8000
12) 21³ = 9261

These are 12 four digit perfect cube numbers  from 1000 to 9999

Total number of 4 digit numbers which are perfect squares and perfect cubes is

68 + 12 = 80

So, number of 4 digit numbers which are neither perfect squares nor perfect cube is 

9000 - 80 = 8920

Answer.

Answer 2

Answer:

8921

Step-by-step explanation:

Number of 4 digit numbers between from 1000 to 9999 (both are included)

= 9999 - 999 = 9000

So, there are 9000 4 digit numbers from 1000 to 9999 (both included)

Perfect square numbers from 1000 to 9999 are as follows.

1) 32² = 1024

2) 33² = 1089

3) 34² = 1156

4) 35² = 1225

5) 36² = 1296

6) 37² = 1369

7) 38² = 1444

8) 39² = 1521

9) 40² = 1600

10) 41² = 1681 

11) 42² = 1764

12) 43² = 1869

13) 44² = 1936

14) 45² = 2025

15) 46² = 2116

16) 47² = 2209

17) 48² = 2304

18) 49² = 2401

19) 50² = 2500

20) 51² = 2601

21) 52² = 2704

22) 53² = 2809

23) 54² = 2916

24) 55² = 3025

25) 56² = 3136

26) 57² = 3249 

27) 58² = 3364

28) 59² = 3481

29) 60² = 3600

30) 61² = 3721

31) 62² = 3844

32) 63² = 3969

33) 64² = 4096

34) 65² = 4225

35) 66² = 4356

36) 67² = 4489

37) 68² = 4624

38) 69² = 4761

39) 70² = 4900

40) 71² = 5041 

41) 72² = 5184 

42) 73² = 5329

43) 74² = 5476

44) 75² = 5625

45) 76² = 5776

46) 77² = 5929

47) 78² = 6084

48) 79² = 6241

49) 80² = 6400

50) 81² = 6521

51) 82² = 6724

52) 83² = 6889

53) 84² = 7056

54) 85² = 7225

55) 86² = 7396

56) 87² = 7569

57) 88² = 7744

58) 89² = 7921

59) 90² = 8100

60) 91² = 8281

61) 92² = 8464

62) 93² = 8649

63) 94² = 8836

64) 95² = 9025

65) 96² = 9216

66) 97² = 9409

67) 98² = 9604

99) 99² = 9801

These are 68 four digit perfect square numbers from 1000 to 9999

Perfect cube numbers from 1000 to 9999 are as follows.

1) 10³ = 1000

2) 11³ = 1331

3) 12³ = 1728

4) 13³ = 2197

5) 14³ = 2744

6) 15³ = 3375

7) 16³ = 4096

8) 17³ = 4913

9) 18³ = 5832

10) 19³ = 6859

11) 20³ = 8000

12) 21³ = 9261

These are 12 four digit perfect cube numbers  from 1000 to 9999

Total number of 4 digit numbers which are perfect squares and perfect cubes is

68 + 12 = 80

Between 10 and 21, we have 16 which is perfect square implying that        16³ = 4096   will also be a perfect square-implying that 4096 occurs in both the groups

68+12-1 = 79

So, number of 4 digit numbers which are neither perfect squares nor perfect cube is

9000-79 = 8921 ---------------------------------------------- Answer