help

This is how I would personally approach this problem.

 

#1: List all the integers from 1 to 100.

 

1	2	3	4	5	6	7	8	9	10
11	12	13	14	15	16	17	18	19	20
21	22	23	24	25	26	27	28	29	30
31	32	33	34	35	36	37	38	39	40
41	42	43	44	45	46	47	48	49	50
51	52	53	54	55	56	57	58	59	60
61	62	63	64	65	66	67	68	69	70
71	72	73	74	75	76	77	78	79	80
81	82	83	84	85	86	87	88	89	90
91	92	93	94	95	96	97	98	99	100

 

#2: Eliminate all even numbers and 1 and 100 from the list since we are only interested in "square-free" numbers bounded between 1 and 100.

 

		3		5		7		9
11		13		15		17		19
21		23		25		27		29
31		33		35		37		39
41		43		45		47		49
51		53		55		57		59
61		63		65		67		69
71		73		75		77		79
81		83		85		87		89
91		93		95		97		99

 

#3: Eliminate multiples of the first perfect square I will test, 3^2, or 9. I am not testing 2^2, or 4 because I have already eliminate the even numbers from the list, so no multiples of 4 remain in the list right now. 

 

		3		5		7
11		13		15		17		19
21		23		25				29
31		33		35		37		39
41		43				47		49
51		53		55		57		59
61				65		67		69
71		73		75		77		79
		83		85		87		89
91		93		95		97

 

#4: Eliminate the multiples of the second perfect square I will test, 5^2, or 25. Again, I am not testing 4^2, or 16, because even numbers have already been filtered out. 

 

		3		5		7
11		13		15		17		19
21		23						29
31		33		35		37		39
41		43				47		49
51		53		55		57		59
61				65		67		69
71		73				77		79
		83		85		87		89
91		93		95		97

 

#5: Eliminate the multiples of 7^2, or 49. Only one number gets eliminated.

 

		3		5		7
11		13		15		17		19
21		23						29
31		33		35		37		39
41		43				47
51		53		55		57		59
61				65		67		69
71		73				77		79
		83		85		87		89
91		93		95		97

 

#6: Normally, I would eliminate the multiples of 9^2, but I already sifted out the multiples of nine, so all the multiples of 9^2 have already been eliminated from the list.

 

#7: There is no need to eliminate the multiples of 11^2, or 121, since that multiple is larger than the largest number we are testing, so we can stop the process here. The numbers that remain are "square-free."

 

In a list, they are 3,5,7,11,13,15,17,19,21,23,29,31,33,35,37,39,41,43,47,51,53,55,57,59,61,65,67,69,71,73,77,79,83,85,87,89,91,93,95,97.