Finding the Least Number Doubled Divisible by 7, 12, and 15

The question asks for the least number which, when doubled, is perfectly divisible by three given numbers: 7, 12, and 15.

If a number is perfectly divisible by 7, 12, and 15, it must be a common multiple of these three numbers. The least such number is the Least Common Multiple (LCM) of 7, 12, and 15.

Let the required least number be \(x\). According to the problem, when \(x\) is doubled, the result (\(2x\)) is perfectly divisible by 7, 12, and 15. This means that \(2x\) must be a multiple of the LCM of 7, 12, and 15.

Understanding Divisibility and Least Common Multiple (LCM)

Divisibility means that when one number is divided by another, the remainder is zero. A common multiple of a set of numbers is a number that is a multiple of every number in the set. The Least Common Multiple (LCM) is the smallest positive number that is a common multiple of two or more numbers.

Calculating the LCM of 7, 12, and 15

To find the LCM, we can use the prime factorization method. We find the prime factors of each number:

• Prime factors of 7: \(7 = 7^1\)
• Prime factors of 12: \(12 = 2 \times 2 \times 3 = 2^2 \times 3^1\)
• Prime factors of 15: \(15 = 3 \times 5 = 3^1 \times 5^1\)

To find the LCM, we take the highest power of all the prime factors that appear in any of the numbers:

The prime factors involved are 2, 3, 5, and 7.

• Highest power of 2: \(2^2\) (from 12)
• Highest power of 3: \(3^1\) (from 12 and 15)
• Highest power of 5: \(5^1\) (from 15)
• Highest power of 7: \(7^1\) (from 7)

LCM(7, 12, 15) = \(2^2 \times 3^1 \times 5^1 \times 7^1 = 4 \times 3 \times 5 \times 7\)

LCM(7, 12, 15) = \(12 \times 35 = 420\)

Solving for the Least Number

As established earlier, \(2x\) must be a multiple of LCM(7, 12, 15). To find the least such number \(x\), \(2x\) must be the least multiple of LCM(7, 12, 15), which is LCM(7, 12, 15) itself.

So, we have the equation: \(2x = \text{LCM}(7, 12, 15)\) \(2x = 420\)

Now, we solve for \(x\) by dividing both sides by 2: \(x = \frac{420}{2}\) \(x = 210\)

Thus, the least number is 210.

Verifying the Solution

Let's check if doubling 210 is perfectly divisible by 7, 12, and 15.

Doubling 210 gives \(2 \times 210 = 420\).

• Is 420 divisible by 7? \(420 \div 7 = 60\). Yes.
• Is 420 divisible by 12? \(420 \div 12 = 35\). Yes.
• Is 420 divisible by 15? \(420 \div 15 = 28\). Yes.

Since 420 is perfectly divisible by 7, 12, and 15, our calculated number, 210, is indeed the least number that satisfies the condition.

Revision Table: Key Concepts

Additional Information: Number Properties

Understanding multiples, factors, and LCM is fundamental in number theory. Related concepts include:

• Factors: Numbers that divide a given number evenly. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12.
• Common Factors: Factors that two or more numbers share. For example, common factors of 12 and 18 are 1, 2, 3, and 6.
• Highest Common Factor (HCF) or Greatest Common Divisor (GCD): The largest of the common factors. For example, HCF(12, 18) = 6.

LCM is particularly useful when dealing with problems involving cycles, events that repeat at regular intervals, or finding the smallest quantity that can be divided equally into different-sized groups. HCF is useful when dividing quantities into the largest possible equal parts.