What is a factor of 6

Happy birthday, Kathy! I hope your day is wonderful. You have grown into a beautiful, talented, prayerful, intelligent, hard-working, and loving young woman. I am grateful you are my daughter. So for your birthday today and for this blog, I’ve created three special puzzles: the first is a birthday cake to celebrate your happy day. To highlight your love of music, the second puzzle is a quarter note. The third puzzle is either a violin, a guitar, or a ukulele, you decide. I love listening as you sing or as you play any of those instruments or the piano. Today for your birthday I will also cut down a tree and make yet another cake with two birthday candles on top in this blog post. So have a fun birthday, today. I love you.

Índice Show

Factor Trees vs. Factor Cakes:
Factors of the Number 6:
A Sum-Difference Puzzle Featuring the Number 6 and its Factors:

Click 12 Factors 2013-11-14 for more puzzles.

Factor Trees vs. Factor Cakes:

What did I mean by cutting down a tree and making yet another cake? Today I will discuss two methods for finding the prime factors of a whole number. One method is making a factor tree and the other is the cake method. To factor a number means to write it as the product of two or more factors. When those two or more factors are all prime factors, it is called a prime factorization of the number. A composite number always has more than two factors. A prime number always has exactly 2 factors, 1 and itself. (ZERO and ONE are neither prime or composite numbers.) Usually, to find the prime factors of a number, a person will usually make a factor tree. The following example shows how this is done:

From this example, you can certainly understand why this algorithm is called a factor tree. It looks exactly like a perfectly-shaped evergreen tree. The problem is that a factor tree doesn’t always look so neat and trim. Here is a factor tree that even Charlie Brown wouldn’t choose:

720 isn’t even that big of a number, but gathering all of the prime numbers from the factor tree and putting them in numerical order would be like picking up a bunch of scattered leaves. It would be like doing . . . yard work. Imagine if you had a number that had many more factors. If one or two of the factors gets lost in the mess, your answer wouldn’t be correct. Notice that some of the prime factors of 720 (2,2,2,2,3,3,5) are not as easy to see as others on the factor tree. That is why I want to chop down that tree. Even if you like to do yard work, do you really want to deal with that big of a mess, . . . especially when you can have cake instead? Look, the cake method is so much more pleasing to the eye, and it is simply an extension of the very familiar division algorithm:

With the cake method, the more factors you have, the bigger the cake will be, but it will always be neatly organized with all the factors on the outside of the cake. And if the largest prime factor of your given number is eleven, you will also have two candles on top of your cake! I find using the cake method to be much less confusing than using a factor tree. Yes, finding prime factors can actually be a piece of cake. The only disadvantage to the cake method is that since you work from the bottom up you have to leave enough space for the cake to rise.

Still, in spite of my opinion, it is best to use whichever method you are more comfortable with.

Now if your appetite for cake has not been satisfied, click on one of the links below for a nice variety of cakes shared by other bloggers.

Another Surprise Rainbow Cake (seraphinacakes.wordpress.com)
The Case for Cake (benmo45.wordpress.com)
“Birthday cake” (smilingmushrooms.wordpress.com)
In Search of The Best Chocolate Cake (lavaletteerica.wordpress.com)

Factors of the Number 6:

6 is a composite number. 6 = 1 x 6 or 2 x 3. Factors of 6: 1, 2, 3, 6. Prime factorization: 2 x 3.

When 6 is a clue in the FIND THE FACTORS puzzle, the pair that will work for that particular puzzle might be 1 x 6, or it might be 2 x 3.

A Sum-Difference Puzzle Featuring the Number 6 and its Factors:

Look at the factor pair puzzle above. Perhaps you will notice that 2 + 3 = 5 and 6 – 1 = 5.

Those are the facts you need to complete the Sum-Difference puzzle below.

Here are the factors (not including negatives), and some multiples, for 1 to 100:

Factors		Multiples
1	1	2	3	4	5	6	7	8	9	10
1, 2	2	4	6	8	10	12	14	16	18	20
1, 3	3	6	9	12	15	18	21	24	27	30
1, 2, 4	4	8	12	16	20	24	28	32	36	40
1, 5	5	10	15	20	25	30	35	40	45	50
1, 2, 3, 6	6	12	18	24	30	36	42	48	54	60
1, 7	7	14	21	28	35	42	49	56	63	70
1, 2, 4, 8	8	16	24	32	40	48	56	64	72	80
1, 3, 9	9	18	27	36	45	54	63	72	81	90
1, 2, 5, 10	10	20	30	40	50	60	70	80	90	100
1, 11	11	22	33	44	55	66	77	88	99	110
1, 2, 3, 4, 6, 12	12	24	36	48	60	72	84	96	108	120
1, 13	13	26	39	52	65	78	91	104	117	130
1, 2, 7, 14	14	28	42	56	70	84	98	112	126	140
1, 3, 5, 15	15	30	45	60	75	90	105	120	135	150
1, 2, 4, 8, 16	16	32	48	64	80	96	112	128	144	160
1, 17	17	34	51	68	85	102	119	136	153	170
1, 2, 3, 6, 9, 18	18	36	54	72	90	108	126	144	162	180
1, 19	19	38	57	76	95	114	133	152	171	190
1, 2, 4, 5, 10, 20	20	40	60	80	100	120	140	160	180	200
1, 3, 7, 21	21	42	63	84	105	126	147	168	189	210
1, 2, 11, 22	22	44	66	88	110	132	154	176	198	220
1, 23	23	46	69	92	115	138	161	184	207	230
1, 2, 3, 4, 6, 8, 12, 24	24	48	72	96	120	144	168	192	216	240
1, 5, 25	25	50	75	100	125	150	175	200	225	250
1, 2, 13, 26	26	52	78	104	130	156	182	208	234	260
1, 3, 9, 27	27	54	81	108	135	162	189	216	243	270
1, 2, 4, 7, 14, 28	28	56	84	112	140	168	196	224	252	280
1, 29	29	58	87	116	145	174	203	232	261	290
1, 2, 3, 5, 6, 10, 15, 30	30	60	90	120	150	180	210	240	270	300
1, 31	31	62	93	124	155	186	217	248	279	310
1, 2, 4, 8, 16, 32	32	64	96	128	160	192	224	256	288	320
1, 3, 11, 33	33	66	99	132	165	198	231	264	297	330
1, 2, 17, 34	34	68	102	136	170	204	238	272	306	340
1, 5, 7, 35	35	70	105	140	175	210	245	280	315	350
1, 2, 3, 4, 6, 9, 12, 18, 36	36	72	108	144	180	216	252	288	324	360
1, 37	37	74	111	148	185	222	259	296	333	370
1, 2, 19, 38	38	76	114	152	190	228	266	304	342	380
1, 3, 13, 39	39	78	117	156	195	234	273	312	351	390
1, 2, 4, 5, 8, 10, 20, 40	40	80	120	160	200	240	280	320	360	400
1, 41	41	82	123	164	205	246	287	328	369	410
1, 2, 3, 6, 7, 14, 21, 42	42	84	126	168	210	252	294	336	378	420
1, 43	43	86	129	172	215	258	301	344	387	430
1, 2, 4, 11, 22, 44	44	88	132	176	220	264	308	352	396	440
1, 3, 5, 9, 15, 45	45	90	135	180	225	270	315	360	405	450
1, 2, 23, 46	46	92	138	184	230	276	322	368	414	460
1, 47	47	94	141	188	235	282	329	376	423	470
1, 2, 3, 4, 6, 8, 12, 16, 24, 48	48	96	144	192	240	288	336	384	432	480
1, 7, 49	49	98	147	196	245	294	343	392	441	490
1, 2, 5, 10, 25, 50	50	100	150	200	250	300	350	400	450	500
1, 3, 17, 51	51	102	153	204	255	306	357	408	459	510
1, 2, 4, 13, 26, 52	52	104	156	208	260	312	364	416	468	520
1, 53	53	106	159	212	265	318	371	424	477	530
1, 2, 3, 6, 9, 18, 27, 54	54	108	162	216	270	324	378	432	486	540
1, 5, 11, 55	55	110	165	220	275	330	385	440	495	550
1, 2, 4, 7, 8, 14, 28, 56	56	112	168	224	280	336	392	448	504	560
1, 3, 19, 57	57	114	171	228	285	342	399	456	513	570
1, 2, 29, 58	58	116	174	232	290	348	406	464	522	580
1, 59	59	118	177	236	295	354	413	472	531	590
1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60	60	120	180	240	300	360	420	480	540	600
1, 61	61	122	183	244	305	366	427	488	549	610
1, 2, 31, 62	62	124	186	248	310	372	434	496	558	620
1, 3, 7, 9, 21, 63	63	126	189	252	315	378	441	504	567	630
1, 2, 4, 8, 16, 32, 64	64	128	192	256	320	384	448	512	576	640
1, 5, 13, 65	65	130	195	260	325	390	455	520	585	650
1, 2, 3, 6, 11, 22, 33, 66	66	132	198	264	330	396	462	528	594	660
1, 67	67	134	201	268	335	402	469	536	603	670
1, 2, 4, 17, 34, 68	68	136	204	272	340	408	476	544	612	680
1, 3, 23, 69	69	138	207	276	345	414	483	552	621	690
1, 2, 5, 7, 10, 14, 35, 70	70	140	210	280	350	420	490	560	630	700
1, 71	71	142	213	284	355	426	497	568	639	710
1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72	72	144	216	288	360	432	504	576	648	720
1, 73	73	146	219	292	365	438	511	584	657	730
1, 2, 37, 74	74	148	222	296	370	444	518	592	666	740
1, 3, 5, 15, 25, 75	75	150	225	300	375	450	525	600	675	750
1, 2, 4, 19, 38, 76	76	152	228	304	380	456	532	608	684	760
1, 7, 11, 77	77	154	231	308	385	462	539	616	693	770
1, 2, 3, 6, 13, 26, 39, 78	78	156	234	312	390	468	546	624	702	780
1, 79	79	158	237	316	395	474	553	632	711	790
1, 2, 4, 5, 8, 10, 16, 20, 40, 80	80	160	240	320	400	480	560	640	720	800
1, 3, 9, 27, 81	81	162	243	324	405	486	567	648	729	810
1, 2, 41, 82	82	164	246	328	410	492	574	656	738	820
1, 83	83	166	249	332	415	498	581	664	747	830
1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84	84	168	252	336	420	504	588	672	756	840
1, 5, 17, 85	85	170	255	340	425	510	595	680	765	850
1, 2, 43, 86	86	172	258	344	430	516	602	688	774	860
1, 3, 29, 87	87	174	261	348	435	522	609	696	783	870
1, 2, 4, 8, 11, 22, 44, 88	88	176	264	352	440	528	616	704	792	880
1, 89	89	178	267	356	445	534	623	712	801	890
1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90	90	180	270	360	450	540	630	720	810	900
1, 7, 13, 91	91	182	273	364	455	546	637	728	819	910
1, 2, 4, 23, 46, 92	92	184	276	368	460	552	644	736	828	920
1, 3, 31, 93	93	186	279	372	465	558	651	744	837	930
1, 2, 47, 94	94	188	282	376	470	564	658	752	846	940
1, 5, 19, 95	95	190	285	380	475	570	665	760	855	950
1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96	96	192	288	384	480	576	672	768	864	960
1, 97	97	194	291	388	485	582	679	776	873	970
1, 2, 7, 14, 49, 98	98	196	294	392	490	588	686	784	882	980
1, 3, 9, 11, 33, 99	99	198	297	396	495	594	693	792	891	990
1, 2, 4, 5, 10, 20, 25, 50, 100	100	200	300	400	500	600	700	800	900	1000