Problem 23 - Fundamentals of Mathematics

Question

Write all the factors of 36 .

Step-by-Step Solution

Answer

The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36.

Step 1: Understand the Concept of Factors

A factor of a number is an integer that divides the number without leaving a remainder. For instance, if we say that a number \( x \) is a factor of 36, then 36 divided by \( x \) should result in an integer.

Step 2: Start with the Smallest Factor

The smallest factor of any number is always 1, because any number divided by 1 is the number itself. So, 1 is a factor of 36.

Step 3: Find the Pair Factor Using Division

To find the next factor, divide 36 by 2. Since 36 divided by 2 equals 18, both 2 and 18 are factors of 36.

Step 4: Continue Finding Pairs

Next, divide 36 by 3. Since 36 divided by 3 equals 12, both 3 and 12 are factors of 36. Repeat this for 4, as 36 divided by 4 equals 9, so 4 and 9 are also factors.

Step 5: Consider the Square Root

Since 36 is a perfect square, its square root, 6, is also a factor. Verify that 6 times 6 gives 36.

Step 6: List All Factors Including Repeated Ones

Ensure to list out all the factors found in steps 2 to 5. The complete set of factors for 36 is: 1, 2, 3, 4, 6, 9, 12, 18, and 36.

Key Concepts

Understanding Integer Division in Factorization

Integer division is crucial for finding factors because it helps determine if one number can be divided by another without leaving a remainder. When you divide two integers and the result is also an integer, it means that the divisor is a factor of the dividend. For example, when we divide 36 by any of its factors, such as 2 or 3, the quotient is an integer, confirming that these are indeed factors.
It works like this:

• 36 divided by 2 equals 18, with no remainder, so 2 is a factor.
• 36 divided by 3 equals 12, also with no remainder, confirming 3 as a factor.

In finding factors, start with the smallest integers and work your way up. This ensures that all possible factors are checked. It's simple, just divide the number by consecutive integers starting from 1, checking each time if the quotient is an integer. This logical approach guarantees you don't miss any potential factors.

The Importance of Perfect Squares

A perfect square is a number that can be expressed as the product of an integer with itself. In terms of factorization, knowing if a number is a perfect square can simplify finding all its factors. For 36, recognizing it as a perfect square is beneficial because \[36 = 6 \times 6\].
This means:

• 6 is a factor that appears twice, making it a crucial consideration for complete factor lists.
• Understanding square roots can quickly reveal a central factor in the list.

Perfect squares are unique because their square root is also a factor. It’s a midpoint in the factor list, dividing smaller factors from larger ones. Knowing the concept of perfect squares not only simplifies calculations but also helps in knowing when to stop direct calculations when listing factors. The symmetry in a perfect square's factors aids in quicker, more efficient factorization for other numbers too.

Simple Pair Factorization

Pair factorization involves finding two numbers that multiply together to give the original number. It's a quick way to list factors because it involves only simple multiplication comparisons. For instance, dividing 36 by smaller factors like 1, 2, 3, 4, helps find their pair counterparts, such as 36, 18, 12, 9 respectively.
Here's the step-by-step:

• Start with small numbers: 1 \( \times \) 36, 2 \( \times \) 18.
• Move to larger factors: 3 \( \times \) 12, 4 \( \times \) 9.
• The perfect square midpoint: 6 \( \times \) 6.

This method ensures all factors are found because each divisor has a partner that multiplies with it to restore the original number. It's effective and visual; seeing numbers in pairs provides clarity. Pair factorization is especially useful because it not only confirms existing factors but also efficiently expands the list ensuring comprehensivity.