What is the least number which when divided by 8 12 15 and 20 leaves in each case a reminder of 5?

Answer

Hint:Least common multiple or smallest common multiple (denoted as L.C.M) of two integers (p and q) is the smallest integer that is divisible by both the integers p and q. We can easily find the L.C.M of given integers simply by using the prime factorization method.

Complete step by step solution:

Therefore, the required smallest number is \[120 + 5 = 125\]

Note:

LCM of 8, 12, 15, and 20 is the smallest number among all common multiples of 8, 12, 15, and 20. The first few multiples of 8, 12, 15, and 20 are (8, 16, 24, 32, 40 . . .), (12, 24, 36, 48, 60 . . .), (15, 30, 45, 60, 75 . . .), and (20, 40, 60, 80, 100 . . .) respectively. There are 3 commonly used methods to find LCM of 8, 12, 15, 20 - by division method, by prime factorization, and by listing multiples.

What is the LCM of 8, 12, 15, and 20?

Answer: LCM of 8, 12, 15, and 20 is 120.

Explanation:

The LCM of four non-zero integers, a(8), b(12), c(15), and d(20), is the smallest positive integer m(120) that is divisible by a(8), b(12), c(15), and d(20) without any remainder.

Methods to Find LCM of 8, 12, 15, and 20

The methods to find the LCM of 8, 12, 15, and 20 are explained below.

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

LCM of 8, 12, 15, and 20 by Prime Factorization

Prime factorization of 8, 12, 15, and 20 is (2 × 2 × 2) = 23, (2 × 2 × 3) = 22 × 31, (3 × 5) = 31 × 51, and (2 × 2 × 5) = 22 × 51 respectively. LCM of 8, 12, 15, and 20 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 23 × 31 × 51 = 120.
Hence, the LCM of 8, 12, 15, and 20 by prime factorization is 120.

LCM of 8, 12, 15, and 20 by Division Method

To calculate the LCM of 8, 12, 15, and 20 by the division method, we will divide the numbers(8, 12, 15, 20) by their prime factors (preferably common). The product of these divisors gives the LCM of 8, 12, 15, and 20.

• Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 8, 12, 15, and 20. Write this prime number(2) on the left of the given numbers(8, 12, 15, and 20), separated as per the ladder arrangement.
• Step 2: If any of the given numbers (8, 12, 15, 20) 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 8, 12, 15, and 20 is the product of all prime numbers on the left, i.e. LCM(8, 12, 15, 20) by division method = 2 × 2 × 2 × 3 × 5 = 120.

LCM of 8, 12, 15, and 20 by Listing Multiples

To calculate the LCM of 8, 12, 15, 20 by listing out the common multiples, we can follow the given below steps:

• Step 1: List a few multiples of 8 (8, 16, 24, 32, 40 . . .), 12 (12, 24, 36, 48, 60 . . .), 15 (15, 30, 45, 60, 75 . . .), and 20 (20, 40, 60, 80, 100 . . .).
• Step 2: The common multiples from the multiples of 8, 12, 15, and 20 are 120, 240, . . .
• Step 3: The smallest common multiple of 8, 12, 15, and 20 is 120.

∴ The least common multiple of 8, 12, 15, and 20 = 120.

☛ Also Check:

LCM of 8, 12, 15, and 20 Examples

1. Example 1: Find the smallest number that is divisible by 8, 12, 15, 20 exactly.

   Solution:

   The smallest number that is divisible by 8, 12, 15, and 20 exactly is their LCM.
   ⇒ Multiples of 8, 12, 15, and 20:

   • Multiples of 8 = 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96, 104, 112, 120, . . . .
   • Multiples of 12 = 12, 24, 36, 48, 60, 72, 84, 96, 108, 120, . . . .
   • Multiples of 15 = 15, 30, 45, 60, 75, 90, 105, 120, . . . .
   • Multiples of 20 = 20, 40, 60, 80, 100, 120, 140, . . . .

   Therefore, the LCM of 8, 12, 15, and 20 is 120.

2. Example 2: Find the smallest number which when divided by 8, 12, 15, and 20 leaves 6 as the remainder in each case.

   Solution:

   The smallest number exactly divisible by 8, 12, 15, and 20 = LCM(8, 12, 15, 20) ⇒ Smallest number which leaves 6 as remainder when divided by 8, 12, 15, and 20 = LCM(8, 12, 15, 20) + 6

   • 8 = 23
   • 12 = 22 × 31
   • 15 = 31 × 51
   • 20 = 22 × 51

   LCM(8, 12, 15, 20) = 23 × 31 × 51 = 120

   ⇒ The required number = 120 + 6 = 126.

3. Example 3: Which of the following is the LCM of 8, 12, 15, 20? 35, 50, 30, 120.

   Solution:

   The value of LCM of 8, 12, 15, and 20 is the smallest common multiple of 8, 12, 15, and 20. The number satisfying the given condition is 120. ∴LCM(8, 12, 15, 20) = 120.

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The LCM of 8, 12, 15, and 20 is 120. To find the LCM (least common multiple) of 8, 12, 15, and 20, we need to find the multiples of 8, 12, 15, and 20 (multiples of 8 = 8, 16, 24, 32 . . . . 120 . . . . ; multiples of 12 = 12, 24, 36, 48 . . . . 120 . . . . ; multiples of 15 = 15, 30, 45, 60 . . . . 120 . . . . ; multiples of 20 = 20, 40, 60, 80, 120 . . . .) and choose the smallest multiple that is exactly divisible by 8, 12, 15, and 20, i.e., 120.

How to Find the LCM of 8, 12, 15, and 20 by Prime Factorization?

To find the LCM of 8, 12, 15, and 20 using prime factorization, we will find the prime factors, (8 = 23), (12 = 22 × 31), (15 = 31 × 51), and (20 = 22 × 51). LCM of 8, 12, 15, and 20 is the product of prime factors raised to their respective highest exponent among the numbers 8, 12, 15, and 20.
⇒ LCM of 8, 12, 15, 20 = 23 × 31 × 51 = 120.

What are the Methods to Find LCM of 8, 12, 15, 20?

The commonly used methods to find the LCM of 8, 12, 15, 20 are:

• Prime Factorization Method
• Listing Multiples
• Division Method

Which of the following is the LCM of 8, 12, 15, and 20? 105, 120, 12, 50

The value of LCM of 8, 12, 15, 20 is the smallest common multiple of 8, 12, 15, and 20. The number satisfying the given condition is 120.