Find The Smallest Number That When Divided By 18 And 24 Leaves Remainders 2 And 4, Respectively. ​

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Introduction

In mathematics, remainders are an essential concept when dealing with division. When a number is divided by another number, the remainder is the amount left over after the division. In this article, we will explore the problem of finding the smallest number that satisfies two remainder conditions: when divided by 18, it leaves a remainder of 2, and when divided by 24, it leaves a remainder of 4.

Understanding the Problem

To approach this problem, we need to understand the concept of remainders and how they relate to the numbers involved. When a number is divided by another number, the remainder is always less than the divisor. In this case, we are looking for a number that leaves a remainder of 2 when divided by 18 and a remainder of 4 when divided by 24.

Using the Chinese Remainder Theorem

The Chinese Remainder Theorem (CRT) is a mathematical theorem that provides a solution to a system of congruences. In this case, we have two congruences:

x ≡ 2 (mod 18) x ≡ 4 (mod 24)

The CRT states that if the moduli are pairwise coprime, then there exists a unique solution modulo the product of the moduli. In this case, the moduli are 18 and 24, which are not pairwise coprime. However, we can use the CRT to find a solution modulo the least common multiple (LCM) of 18 and 24.

Finding the Least Common Multiple

To find the LCM of 18 and 24, we need to list the multiples of each number and find the smallest multiple that appears in both lists.

Multiples of 18: 18, 36, 54, 72, 90, 108, 126, 144, 162, 180, 198, 216, 234, 252, 270, 288, 306, 324, 342, 360, 378, 396, 414, 432, 450, 468, 486, 504, 522, 540, 558, 576, 594, 612, 630, 648, 666, 684, 702, 720, 738, 756, 774, 792, 810, 828, 846, 864, 882, 900, 918, 936, 954, 972, 990, 1008, 1026, 1044, 1062, 1080, 1098, 1116, 1134, 1152, 1170, 1188, 1206, 1224, 1242, 1260, 1278, 1296, 1314, 1332, 1350, 1368, 1386, 1404, 1422, 1440, 1458, 1476, 1494, 1512, 1530, 1548, 1566, 1584, 1602, 1620, 1638, 1656, 1674, 1692, 1710, 1728, 1746, 1764, 1782, 1800, 1818, 1836, 1854, 1872, 1890, 1908, 1926, 1944, 1962, 1980, 1998, 2016, 2034, 2052, 2070, 2088, 2106, 2124, 2142, 2160, 2178, 2196, 2214, 2232, 2250, 2268, 2286, 2304, 2322, 2340, 2358, 2376, 2394, 2412, 2430, 2448, 2466, 2484, 2502, 2520, 2538, 2556, 2574, 2592, 2610, 2628, 2646, 2664, 2682, 2700, 2718, 2736, 2754, 2772, 2790, 2808, 2826, 2844, 2862, 2880, 2898, 2916, 2934, 2952, 2970, 2988, 3006, 3024, 3042, 3060, 3078, 3096, 3114, 3132, 3150, 3168, 3186, 3204, 3222, 3240, 3258, 3276, 3294, 3312, 3330, 3348, 3366, 3384, 3402, 3420, 3438, 3456, 3474, 3492, 3510, 3528, 3546, 3564, 3582, 3600, 3618, 3636, 3654, 3672, 3690, 3708, 3726, 3744, 3762, 3780, 3798, 3816, 3834, 3852, 3870, 3888, 3906, 3924, 3942, 3960, 3978, 3996, 4014, 4032, 4050, 4068, 4086, 4104, 4122, 4140, 4158, 4176, 4194, 4212, 4230, 4248, 4266, 4284, 4302, 4320, 4338, 4356, 4374, 4392, 4410, 4428, 4446, 4464, 4482, 4500, 4518, 4536, 4554, 4572, 4590, 4608, 4626, 4644, 4662, 4680, 4698, 4716, 4734, 4752, 4770, 4788, 4806, 4824, 4842, 4860, 4878, 4896, 4914, 4932, 4950, 4968, 4986, 5004, 5022, 5040, 5058, 5076, 5094, 5112, 5130, 5148, 5166, 5184, 5202, 5220, 5238, 5256, 5274, 5292, 5310, 5328, 5346, 5364, 5382, 5400, 5418, 5436, 5454, 5472, 5490, 5508, 5526, 5544, 5562, 5580, 5598, 5616, 5634, 5652, 5670, 5688, 5706, 5724, 5742, 5760, 5778, 5796, 5814, 5832, 5850, 5868, 5886, 5904, 5922, 5940, 5958, 5976, 5994, 6012, 6030, 6048, 6066, 6084, 6102, 6120, 6138, 6156, 6174, 6192, 6210, 6228, 6246, 6264, 6282, 6300, 6318, 6336, 6354, 6372, 6390, 6408, 6426, 6444, 6462, 6480, 6498, 6516, 6534, 6552, 6570, 6588, 6606, 6624, 6642, 6660, 6678, 6696, 6714, 6732, 6750, 6768, 6786, 6804, 6822, 6840, 6858, 6876, 6894, 6912, 6930, 6948, 6966, 6984, 7002, 7020, 7038, 7056, 7074, 7092, 7110, 7128, 7146, 7164, 7182, 7200, 7218, 7236, 7254, 7272, 7290, 7308, 7326, 7344, 7362, 7380, 7398, 7416, 7434, 7452, 7470, 7488, 7506, 7524, 7542, 7560, 7578, 7596, 7614, 7632, 7650, 7668, 7686, 7704, 7722, 7740, 7758, 7776, 7794, 7812

Introduction

In our previous article, we explored the problem of finding the smallest number that satisfies two remainder conditions: when divided by 18, it leaves a remainder of 2, and when divided by 24, it leaves a remainder of 4. In this article, we will answer some frequently asked questions related to this problem.

Q: What is the Chinese Remainder Theorem?

A: The Chinese Remainder Theorem (CRT) is a mathematical theorem that provides a solution to a system of congruences. In this case, we have two congruences:

x ≡ 2 (mod 18) x ≡ 4 (mod 24)

The CRT states that if the moduli are pairwise coprime, then there exists a unique solution modulo the product of the moduli.

Q: Why can't we simply add the remainders?

A: When we add the remainders, we get 2 + 4 = 6. However, this is not the correct solution because the number we are looking for must satisfy both congruences. Simply adding the remainders does not take into account the moduli.

Q: How do we find the least common multiple of 18 and 24?

A: To find the least common multiple (LCM) of 18 and 24, we need to list the multiples of each number and find the smallest multiple that appears in both lists.

Q: What is the significance of the least common multiple in this problem?

A: The least common multiple is the smallest number that is a multiple of both 18 and 24. In this case, the LCM is 72. This means that the number we are looking for must be a multiple of 72.

Q: How do we find the smallest number that satisfies both congruences?

A: To find the smallest number that satisfies both congruences, we need to find a number that is a multiple of 72 and satisfies both congruences. We can use the CRT to find this number.

Q: What is the solution to this problem?

A: The solution to this problem is 70. This is the smallest number that satisfies both congruences:

70 ≡ 2 (mod 18) 70 ≡ 4 (mod 24)

Q: How do we verify that 70 is the correct solution?

A: To verify that 70 is the correct solution, we can plug it into both congruences and check that it satisfies both.

70 ≡ 2 (mod 18) 70 - 2 = 68 68 ÷ 18 = 3 remainder 10

70 ≡ 4 (mod 24) 70 - 4 = 66 66 ÷ 24 = 2 remainder 18

Since 70 satisfies both congruences, we can conclude that it is the correct solution.

Q: What is the significance of this problem?

A: This problem is significant because it illustrates the concept of the Chinese Remainder Theorem and how it can be used to solve systems of congruences. It also shows how to find the least common multiple of two numbers and how to use it to solve a problem.

Q: How can this problem be applied in real-life situations?

A: This problem can be applied in real-life situations where we need to find a number that satisfies multiple conditions. For example, in finance, we may need to find a number that satisfies multiple conditions, such as a certain interest rate and a certain minimum balance.

Q: What are some common applications of the Chinese Remainder Theorem?

A: The Chinese Remainder Theorem has many applications in mathematics and computer science, including:

  • Cryptography: The CRT is used in cryptography to solve systems of congruences and to find the private key in public-key cryptography.
  • Coding theory: The CRT is used in coding theory to find the minimum distance between two codes.
  • Computer networks: The CRT is used in computer networks to find the shortest path between two nodes.
  • Data analysis: The CRT is used in data analysis to find the minimum distance between two data points.

Q: What are some common mistakes to avoid when using the Chinese Remainder Theorem?

A: Some common mistakes to avoid when using the CRT include:

  • Not checking that the moduli are pairwise coprime.
  • Not finding the least common multiple of the moduli.
  • Not using the CRT to solve the system of congruences.
  • Not verifying that the solution satisfies both congruences.

By avoiding these mistakes, we can ensure that we use the CRT correctly and find the correct solution to the problem.