Finding the Greatest Three-Digit Number Divisible by Multiple Numbers

The question asks for the largest three-digit number that is perfectly divisible by 14, 28, and 42. A number that is divisible by multiple numbers must also be divisible by their Least Common Multiple (LCM). Therefore, the first step is to find the LCM of 14, 28, and 42.

Calculating the Least Common Multiple (LCM)

To find the LCM, we can use the prime factorization method.

• Prime factors of 14: \(14 = 2 \times 7\)
• Prime factors of 28: \(28 = 2 \times 2 \times 7 = 2^2 \times 7\)
• Prime factors of 42: \(42 = 2 \times 3 \times 7\)

The LCM is found by taking the highest power of all prime factors present in the numbers.

\(LCM(14, 28, 42) = 2^{\text{highest power}} \times 3^{\text{highest power}} \times 7^{\text{highest power}}\)

• Highest power of 2 is \(2^2\) (from 28)
• Highest power of 3 is \(3^1\) (from 42)
• Highest power of 7 is \(7^1\) (from 14, 28, and 42)

So, \(LCM(14, 28, 42) = 2^2 \times 3 \times 7 = 4 \times 3 \times 7 = 12 \times 7 = 84\).

This means any number divisible by 14, 28, and 42 must be a multiple of 84.

Finding the Greatest Three-Digit Multiple

We are looking for the greatest three-digit number that is a multiple of 84. The greatest three-digit number is 999.

To find the greatest multiple of 84 less than or equal to 999, we divide 999 by 84:

\(999 \div 84\)

Using division:

Let's perform the division:

\(999 = 84 \times q + r\)

Where \(q\) is the quotient and \(r\) is the remainder.

\(999 \div 84\)

We can estimate: \(84 \times 10 = 840\). \(999 - 840 = 159\). \(84 \times 1 = 84\). \(159 - 84 = 75\). So, \(999 = 84 \times 11 + 75\). The quotient is 11 and the remainder is 75.

The largest multiple of 84 less than 999 is \(84 \times \text{quotient}\) if the remainder is greater than 0. If the remainder was 0, 999 itself would be the multiple.

Greatest multiple of 84 less than or equal to 999 is \(84 \times 11\).

\(84 \times 11 = 924\).

Verification

The number 924 is a three-digit number. It is a multiple of 84, which means it is divisible by 14, 28, and 42. Any multiple of 84 greater than 924 would be \(84 \times 12 = 1008\), which is a four-digit number. Therefore, 924 is the greatest three-digit number divisible by 14, 28, and 42.

Analyzing the Options

Let's check the given options:

• Option 1: 964. Is 964 divisible by 84? \(964 \div 84 \approx 11.4\). No.
• Option 2: 924. Is 924 divisible by 84? \(924 \div 84 = 11\). Yes.
• Option 3: 934. Is 934 divisible by 84? \(934 \div 84 \approx 11.1\). No.
• Option 4: 942. Is 942 divisible by 84? \(942 \div 84 \approx 11.2\). No.

Only 924 is divisible by 84 (and hence by 14, 28, and 42).

Thus, the greatest three-digit number divisible by 14, 28, and 42 is 924.

Revision Table: Divisibility Concepts

Additional Information: Using LCM in Divisibility Problems

Problems asking for a number divisible by several different numbers require finding the LCM of those numbers. The number you are looking for will always be a multiple of the LCM. If you need the smallest such number, it is the LCM itself (unless constraints like "greater than 1" or "two-digit" are given). If you need the greatest such number within a range, you find the largest multiple of the LCM within that range.

For instance, to find the smallest four-digit number divisible by 14, 28, and 42:

• LCM is 84.
• Smallest four-digit number is 1000.
• Divide 1000 by 84: \(1000 = 84 \times 11 + 76\).
• The smallest multiple of 84 greater than or equal to 1000 is the next multiple after \(84 \times 11\), which is \(84 \times 12\).
• \(84 \times 12 = 1008\). This is the smallest four-digit number divisible by 14, 28, and 42.