Factoring and Multiples

Well, isn't that just a happy little math problem! The greatest common factor of 40 and 42 is 2. You see, both 40 and 42 can be divided by 2 evenly, making it the largest number that can divide them both without any remainders. Just like painting, math can be a beautiful and harmonious experience when you find the common factors that bring everything together.

6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, 84, 90, 96, 102, 108, 114, 120, 126, 132, 138, 144, 150, 156, 162, 168, 174, 180, 186, 192, 198, 204, 210, 216, 222, 228, 234, 240, 246, 252, 258, 264, 270, 276, 282, 288, 294, 300, 306, 312, 318, 324, 330, 336, 342, 348, 354, 360, 366, 372, 378, 384, 390, 396, 402, 408, 414, 420, 426, 432, 438, 444, 450, 456, 462, 468, 474, 480, 486, 492, 498, 504, 510, 516, 522, 528, 534, 540, 546, 552, 558, 564, 570, 576, 582, 588, 594, 600, 606, 612, 618, 624, 630, 636, 642, 648, 654, 660, 666, 672, 678, 684, 690, 696, 702, 708, 714, 720, 726, 732, 738, 744, 750, 756, 762, 768, 774, 780, 786, 792, 798, 804, 810, 816, 822, 828, 834, 840, 846, 852, 858, 864, 870, 876, 882, 888, 894, 900, 906, 912, 918, 924, 930, 936, 942, 948, 954, 960, 966, 972, 978, 984, 990, 996, 1002, 1008, 1014, 1020, 1026, 1032, 1038, 1044, 1050, 1056, 1062, 1068, 1074, 1080, 1086, 1092, 1098, 1104, 1110, 1116, 1122, 1128, 1134, 1140, 1146, 1152, 1158, 1164, 1170, 1176, 1182, 1188, 1194, 1200, 1206, 1212, 1218, 1224, 1230, 1236, 1242, 1248, 1254, 1260, 1266, 1272, 1278, 1284, 1290, 1296, 1302, 1308, 1314, 1320, 1326, 1332, 1338, 1344, 1350, 1356, 1362, 1368, 1374, 1380, 1386, 1392, 1398, 1404, 1410, 1416, 1422, 1428, 1434, 1440, 1446, 1452, 1458, 1464, 1470, 1476, 1482, 1488, 1494, 1500, 1506, 1512, 1518, 1524, 1530, 1536, 1542, 1548, 1554, 1560, 1566, 1572, 1578, 1584, 1590, 1596, 1602, 1608, 1614, 1620, 1626, 1632, 1638, 1644, 1650, 1656, 1662, 1668, 1674, 1680, 1686, 1692, 1698, 1704, 1710, 1716, 1722, 1728, 1734, 1740, 1746, 1752, 1758, 1764, 1770, 1776, 1782, 1788, 1794, 1800, 1806, 1812, 1818, 1824, 1830, 1836, 1842, 1848, 1854, 1860, 1866, 1872, 1878, 1884, 1890, 1896, 1902, 1908, 1914, 1920, 1926, 1932, 1938, 1944, 1950, 1956, 1962, 1968, 1974, 1980, 1986, 1992, 1998.

To find two numbers that multiply to 748, we can start by factoring 748 into its prime factors: 2 x 2 x 11 x 17. From this prime factorization, we can see that 4 x 187 equals 748. Therefore, the two numbers that multiply to 748 are 4 and 187.

The greatest common factor (GCF) of 49, 77, and 91 is 7. To find the GCF, you first need to find the prime factors of each number: 49 = 7 x 7, 77 = 7 x 11, and 91 = 7 x 13. Then, identify the common prime factors among the numbers, which in this case is 7. Therefore, the GCF of 49, 77, and 91 is 7.

The GCF is 7.

Oh, dude, the Highest Common Multiple (HCM) of 35, 77, and 91 is 1. Yeah, you heard me right, just 1. Those numbers don't have any common multiples other than the number 1. So, there you have it, the thrilling world of HCMs.

Well, to get a negative number, when multiplying, you must have a negative and a posative. Therefore their would be one negative and posative solution, which would allow you to find a GCF for both a posative and negative number. Here's and example: Find the GCF of 4, and -12

You would factor each of them : you would have

4: 2,2 1,4

-12: 6,-2 2,-6 12,-1 .12,1 3,-4 4,-3

You would then determine that using that data, 4 would be the greatest common facotr of those two numbers.

Factors of 900 are: 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 25, 30, 36, 45, 50, 60, 75, 90, 100, 150, 180, 225, 300, 450, and 900

180 has 18 (1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 38, 45, 60, 90, 180)

300 has 18 (1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 25, 30, 50, 60, 75, 100, 150, 300)

That's it, 2 of them

9x4 = 32*x4 has (2+1)*(4+1) = 3*5 = 15 different factors.

When an expression is written out as the product of its prime factors in index form, the number of factors can be calculated as the product of each index increased by one.

Oh, dude, the least common multiple of 27 and 45 is 135. It's like the cool kid that doesn't hang out with the other numbers much. They're all like, "Hey, 135, wanna come play with us?" And 135's just like, "Nah, I'm good on my own, thanks."

Oh, that's a fun question! Let's think about it like painting a happy little tree. The first nine common multiples of 3.14 are 3.14, 6.28, 9.42, 12.56, 15.7, 18.84, 21.98, 25.12, and 28.26. Just like adding colors to a canvas, these multiples of 3.14 come together in a beautiful pattern.

Oh, dude, the least common multiple of 75, 90, and 105 is 630. It's like the number that these three can all hang out with without any drama. So, yeah, 630 is the magic number where everyone gets along just fine.

Oh, what a happy little prime factorization we have here! Let's take a closer look. When we multiply 2 by 3 by 3 by 5, we get 90. So, this prime factorization is for the number 90. Isn't that just lovely?

To find the Least Common Multiple (LCM) of 72, 144, and 216, we first need to prime factorize each number.

72 = 2^3 * 3^2 144 = 2^4 * 3^2 216 = 2^3 * 3^3

Then, we take the highest power of each prime factor that appears in any of the numbers. So, the LCM of 72, 144, and 216 is 2^4 * 3^3 = 288.

Oh, isn't that a lovely question! To find the Least Common Multiple (LCM) of 294, 252, and 210, we look for the smallest number that all three can divide into evenly. Start by finding the prime factors of each number and then multiply the highest power of all the factors together. It's like painting a beautiful picture with numbers - just take your time and enjoy the process, my friend.

The factors of 256 are the numbers that can be multiplied together to equal 256. These factors include 1, 2, 4, 8, 16, 32, 64, 128, and 256. Each of these numbers can divide evenly into 256 without leaving a remainder, making them factors of 256.

1, 660, 5580

The LCM of 9, 12 and 36 is 36.

To find the greatest common factor (GCF) of 48, 75, and 120, first factorize each number into its prime factors: 48 = 2^4 * 3, 75 = 3 * 5^2, and 120 = 2^3 * 3 * 5. Then, identify the common prime factors among the numbers, which are 3 and 5. The GCF is the product of these common prime factors, which is 3 * 5 = 15. Therefore, the greatest common factor of 48, 75, and 120 is 15.

To find the prime factors of 490, we start by dividing it by the smallest prime number, which is 2. 490 ÷ 2 = 245. Next, we divide 245 by the next smallest prime number, which is 5. 245 ÷ 5 = 49. Finally, 49 is a square of 7, which is a prime number. Therefore, the prime factors of 490 are 2 x 5 x 7 x 7.

The greatest common factor (GCF) of two numbers is the largest number that divides both of them without leaving a remainder. To find the GCF of 375 and 100, we need to identify the common factors of both numbers. The factors of 375 are 1, 3, 5, 15, 25, 75, and 375, while the factors of 100 are 1, 2, 4, 5, 10, 20, 25, 50, and 100. The greatest common factor of 375 and 100 is 25, as it is the largest number that both 375 and 100 can be divided by evenly.

The common factors of 220 and 284 are the numbers that can evenly divide both 220 and 284 without leaving a remainder. To find the common factors, we first need to find the prime factorization of each number. The prime factorization of 220 is 2^2 * 5 * 11, and the prime factorization of 284 is 2^2 * 71. The common factors of 220 and 284 are the numbers that appear in both factorizations, which are 2 and 4.

The prime factors of 84 are: 2, 3 and 7. So, 2 x 2 x 3 x 7 = 84

Oh, dude, like, the common factors of 9, 21, and 63 are 1 and 3. Yeah, that's it. Not much to it, really. Just a couple of numbers hanging out together, keeping it simple.