Find the smallest number which when divided by 36,45 and 63 gives a remainder 8 in each case

LCM of 21, 28, 36, and 45 is the smallest number among all common multiples of 21, 28, 36, and 45. The first few multiples of 21, 28, 36, and 45 are (21, 42, 63, 84, 105 . . .), (28, 56, 84, 112, 140 . . .), (36, 72, 108, 144, 180 . . .), and (45, 90, 135, 180, 225 . . .) respectively. There are 3 commonly used methods to find LCM of 21, 28, 36, 45 - by prime factorization, by division method, and by listing multiples.

What is the LCM of 21, 28, 36, and 45?

Answer: LCM of 21, 28, 36, and 45 is 1260.

Explanation:

The LCM of four non-zero integers, a(21), b(28), c(36), and d(45), is the smallest positive integer m(1260) that is divisible by a(21), b(28), c(36), and d(45) without any remainder.

Methods to Find LCM of 21, 28, 36, and 45

Let's look at the different methods for finding the LCM of 21, 28, 36, and 45.

• By Prime Factorization Method
• By Division Method
• By Listing Multiples

LCM of 21, 28, 36, and 45 by Prime Factorization

Prime factorization of 21, 28, 36, and 45 is (3 × 7) = 31 × 71, (2 × 2 × 7) = 22 × 71, (2 × 2 × 3 × 3) = 22 × 32, and (3 × 3 × 5) = 32 × 51 respectively. LCM of 21, 28, 36, and 45 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 22 × 32 × 51 × 71 = 1260.
Hence, the LCM of 21, 28, 36, and 45 by prime factorization is 1260.

LCM of 21, 28, 36, and 45 by Division Method

To calculate the LCM of 21, 28, 36, and 45 by the division method, we will divide the numbers(21, 28, 36, 45) by their prime factors (preferably common). The product of these divisors gives the LCM of 21, 28, 36, and 45.

• Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 21, 28, 36, and 45. Write this prime number(2) on the left of the given numbers(21, 28, 36, and 45), separated as per the ladder arrangement.
• Step 2: If any of the given numbers (21, 28, 36, 45) is a multiple of 2, divide it by 2 and write the quotient below it. Bring down any number that is not divisible by the prime number.
• Step 3: Continue the steps until only 1s are left in the last row.

The LCM of 21, 28, 36, and 45 is the product of all prime numbers on the left, i.e. LCM(21, 28, 36, 45) by division method = 2 × 2 × 3 × 3 × 5 × 7 = 1260.

LCM of 21, 28, 36, and 45 by Listing Multiples

To calculate the LCM of 21, 28, 36, 45 by listing out the common multiples, we can follow the given below steps:

• Step 1: List a few multiples of 21 (21, 42, 63, 84, 105 . . .), 28 (28, 56, 84, 112, 140 . . .), 36 (36, 72, 108, 144, 180 . . .), and 45 (45, 90, 135, 180, 225 . . .).
• Step 2: The common multiples from the multiples of 21, 28, 36, and 45 are 1260, 2520, . . .
• Step 3: The smallest common multiple of 21, 28, 36, and 45 is 1260.

∴ The least common multiple of 21, 28, 36, and 45 = 1260.

☛ Also Check:

LCM of 21, 28, 36, and 45 Examples

1. Example 1: Find the smallest number that is divisible by 21, 28, 36, 45 exactly.

   Solution:

   The value of LCM(21, 28, 36, 45) will be the smallest number that is exactly divisible by 21, 28, 36, and 45.
   ⇒ Multiples of 21, 28, 36, and 45:

   • Multiples of 21 = 21, 42, 63, 84, 105, 126, 147, 168, 189, 210, . . . ., 1218, 1239, 1260, . . . .
   • Multiples of 28 = 28, 56, 84, 112, 140, 168, 196, 224, 252, 280, . . . ., 1148, 1176, 1204, 1232, 1260, . . . .
   • Multiples of 36 = 36, 72, 108, 144, 180, 216, 252, 288, 324, 360, . . . ., 1152, 1188, 1224, 1260, . . . .
   • Multiples of 45 = 45, 90, 135, 180, 225, 270, 315, 360, 405, 450, . . . ., 1125, 1170, 1215, 1260, . . . .

   Therefore, the LCM of 21, 28, 36, and 45 is 1260.

2. Example 2: Find the smallest number which when divided by 21, 28, 36, and 45 leaves 3 as the remainder in each case.

   Solution:

   The smallest number exactly divisible by 21, 28, 36, and 45 = LCM(21, 28, 36, 45) ⇒ Smallest number which leaves 3 as remainder when divided by 21, 28, 36, and 45 = LCM(21, 28, 36, 45) + 3

   • 21 = 31 × 71
   • 28 = 22 × 71
   • 36 = 22 × 32
   • 45 = 32 × 51

   LCM(21, 28, 36, 45) = 22 × 32 × 51 × 71 = 1260

   ⇒ The required number = 1260 + 3 = 1263.

3. Example 3: Which of the following is the LCM of 21, 28, 36, 45? 21, 96, 1260, 18.

   Solution:

   The value of LCM of 21, 28, 36, and 45 is the smallest common multiple of 21, 28, 36, and 45. The number satisfying the given condition is 1260. ∴LCM(21, 28, 36, 45) = 1260.

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The LCM of 21, 28, 36, and 45 is 1260. To find the least common multiple (LCM) of 21, 28, 36, and 45, we need to find the multiples of 21, 28, 36, and 45 (multiples of 21 = 21, 42, 63, 84 . . . . 1260 . . . . ; multiples of 28 = 28, 56, 84, 112 . . . . 1260 . . . . ; multiples of 36 = 36, 72, 108, 144 . . . . 1260 . . . . ; multiples of 45 = 45, 90, 135, 180 . . . . 1260 . . . . ) and choose the smallest multiple that is exactly divisible by 21, 28, 36, and 45, i.e., 1260.

Which of the following is the LCM of 21, 28, 36, and 45? 40, 42, 35, 1260

The value of LCM of 21, 28, 36, 45 is the smallest common multiple of 21, 28, 36, and 45. The number satisfying the given condition is 1260.

How to Find the LCM of 21, 28, 36, and 45 by Prime Factorization?

To find the LCM of 21, 28, 36, and 45 using prime factorization, we will find the prime factors, (21 = 31 × 71), (28 = 22 × 71), (36 = 22 × 32), and (45 = 32 × 51). LCM of 21, 28, 36, and 45 is the product of prime factors raised to their respective highest exponent among the numbers 21, 28, 36, and 45.
⇒ LCM of 21, 28, 36, 45 = 22 × 32 × 51 × 71 = 1260.

What are the Methods to Find LCM of 21, 28, 36, 45?

The commonly used methods to find the LCM of 21, 28, 36, 45 are:

• Prime Factorization Method
• Division Method
• Listing Multiples

Answer

Hint: First of all let the least number be N. Then using the division theorem, N = dq + r, write, N = 25a + 9, N = 40b + 9 and N = 60c + 9. Find N by taking LCM of 25, 40 and 9 and adding 9 to it.“Complete step-by-step answer:” Here, we have to find the least number which when divided by 25, 40 and 60 leaves 9 as the remainder in each case. Before solving this question, we must know what division theorem is. Division theorem states that “If ‘n’ is any integer and ‘d’ is a positive integer, there exist unique integers ‘q’ and ‘r’ such that, $n=dq+r$ where 0${\leq}$r<$d$Here, ‘n’ is the number or the dividend, ‘d’ is the divisor, ‘q’ is the quotient and ‘r’ is the remainder.For example, if we divide a number or dividend that is, say 16 by divisor, say 5, we get quotient as 3 and remainder as 1.By division theorem, we can write it as 16 = 5 (2) + 1Now, we have to find the least number which when divided by 25, 40, 60 leaves 9 as the remainder in each case.Here, let us consider the least number to be N. As we know that 25, 40, 60 are divisors and 9 is the remainder in each case. Therefore, by division therefore, we get\[\begin{align}  & N=25a+9....\left( i \right) \\  & N=40b+9....\left( ii \right) \\  & N=60c+9....\left( iii \right) \\ \end{align}\]where a, b, and c are quotients in each case.By subtracting 9 from both sides of the equation (i), (ii) and (iii), we get, \[\begin{align}  & \Rightarrow \left( N-9 \right)=25a \\  & \Rightarrow \left( N-9 \right)=40b \\  & \Rightarrow \left( N-9 \right)=60c \\ \end{align}\]As we know that a, b, and c are integers, therefore we have to find the least value of (N – 9) such that it is a multiple of 25, 40 and 60. That means we have to find the LCM or lowest common multiple of 25, 40 and 60.Now, we will find the LCM of 25, 40 and 60 as follows: