Finding the Smallest Integer Divisible by Multiple Numbers

The question asks for the smallest integer that is divisible by 3, 7, and 18. This is equivalent to finding the Least Common Multiple (LCM) of these three numbers.

The LCM of a set of numbers is the smallest positive integer that is a multiple of all the numbers in the set.

We can find the LCM using the prime factorization method. This involves finding the prime factors of each number and then multiplying the highest power of each prime factor that appears in any of the factorizations.

Step-by-Step Calculation of LCM

Let's find the prime factorization of each number:

• Prime factorization of 3: The number 3 is a prime number itself. So, $3 = 3^1$.
• Prime factorization of 7: The number 7 is also a prime number. So, $7 = 7^1$.
• Prime factorization of 18: We can factorize 18 as follows:
  • $18 = 2 \times 9$
  • $18 = 2 \times 3 \times 3$
  • $18 = 2^1 \times 3^2$

Now, we list all the prime factors that appear in the factorizations of 3, 7, and 18. The prime factors are 2, 3, and 7.

Next, we find the highest power of each of these prime factors that appears in any of the factorizations:

• Highest power of 2: The prime factor 2 appears only in the factorization of 18 as $2^1$. The highest power is $2^1$.
• Highest power of 3: The prime factor 3 appears as $3^1$ in the factorization of 3 and as $3^2$ in the factorization of 18. The highest power is $3^2$.
• Highest power of 7: The prime factor 7 appears only in the factorization of 7 as $7^1$. The highest power is $7^1$.

To find the LCM, we multiply these highest powers together:

$\text{LCM}(3, 7, 18) = 2^1 \times 3^2 \times 7^1$

$\text{LCM}(3, 7, 18) = 2 \times 9 \times 7$

$\text{LCM}(3, 7, 18) = 18 \times 7$

$\text{LCM}(3, 7, 18) = 126$

Verification

Let's check if 126 is divisible by 3, 7, and 18:

• $126 \div 3 = 42$ (126 is divisible by 3)
• $126 \div 7 = 18$ (126 is divisible by 7)
• $126 \div 18 = 7$ (126 is divisible by 18)

Since 126 is divisible by all three numbers and it is the LCM, it is the smallest such integer.

Comparing with Options

Let's look at the given options:

1. 63: Divisible by 3 and 7, but not by 18 ($63 \div 18 = 3.5$).
2. 252: Divisible by 3, 7, and 18, but it is larger than 126.
3. 72: Divisible by 3 and 18, but not by 7 ($72 \div 7 \approx 10.28$).
4. 126: Divisible by 3, 7, and 18.

The smallest integer among the options that is divisible by 3, 7, and 18 is 126.

Revision Table: Prime Factorization and LCM

To find the LCM, take the highest power of each unique prime factor ($2^1, 3^2, 7^1$) and multiply them: $2 \times 9 \times 7 = 126$.

Additional Information: LCM vs. HCF (GCD)

While finding the LCM (Least Common Multiple), we look for the smallest common multiple. There is also a concept called HCF (Highest Common Factor) or GCD (Greatest Common Divisor).

• HCF/GCD: The largest number that divides into two or more numbers without leaving a remainder. To find the HCF, you take the lowest power of the common prime factors.
• For example, the HCF of 12 and 18:
  • $12 = 2^2 \times 3^1$
  • $18 = 2^1 \times 3^2$
  • Common prime factors are 2 and 3.
  • Lowest power of 2 is $2^1$.
  • Lowest power of 3 is $3^1$.
  • HCF(12, 18) = $2^1 \times 3^1 = 6$.
• LCM: The smallest positive integer that is a multiple of all the given numbers. To find the LCM, you take the highest power of all unique prime factors.
• For example, the LCM of 12 and 18:
  • $12 = 2^2 \times 3^1$
  • $18 = 2^1 \times 3^2$
  • Unique prime factors are 2 and 3.
  • Highest power of 2 is $2^2$.
  • Highest power of 3 is $3^2$.
  • LCM(12, 18) = $2^2 \times 3^2 = 4 \times 9 = 36$.

The question asked for the smallest integer divisible by the numbers, which is the definition of the Least Common Multiple (LCM).