Problem 1 - Elementary and Intermediate Algebra

Question

In each of Exercises \(1-4\) match the description with the appropriate number from the list on the right. 1.____ A factor of 35 A. 2 B. 7 C. 60 D. 65

Step-by-Step Solution

Answer

7 is a factor of 35.

Step 1: Understand what a factor is

A factor of a number is a number that divides it exactly without leaving a remainder. For instance, if ‘b’ is a factor of ‘a’, then when ‘a’ is divided by ‘b’, the remainder is 0.

Step 2: List the factors of 35

To find the factors of 35, determine the numbers that can divide 35 without leaving a remainder. The factors of 35 are 1, 5, 7, and 35.

Step 3: Match the correct number

From the list provided (2, 7, 60, 65), identify the number that is a factor of 35. Checking each: 2 (no), 7 (yes), 60 (no), 65 (no). The correct number is 7.

Key Concepts

Division

In mathematics, division is one of the four fundamental arithmetic operations. It involves dividing a number (the dividend) by another number (the divisor) to obtain a quotient. For example, in the case of 35 divided by 7, 35 is the dividend and 7 is the divisor, leading us to the quotient, which is 5. When you divide and get no remainder, it means the divisor is a factor of the dividend. This is why understanding division is crucial when determining factors.
If you found the explanation about factors confusing, recall: if a number **'a'** can be divided by another number **'b'** without any remainder, then **'b'** is a factor of **'a'**. In simpler terms, factoring is just a specific application of division.

Remainders

When performing division, sometimes the numbers don’t divide evenly, and you’re left with a number that can’t be further divided by your divisor. This leftover is known as the remainder. For example, dividing 35 by 6, you perform the division like this:
- 35 divided by 6 is 5 (the closest you can get without exceeding 35).
- 5 times 6 is 30.
- 35 minus 30 leaves you 5.
Thus, 35 divided by 6 leaves a quotient of 5 and a remainder of 5.
When checking if one number is a factor of another, the aim is to see if the remainder is zero upon division. If it isn’t, it means that number doesn’t exactly divide the other and hence isn’t a factor.

Multiples

Multiples of a number are produced by multiplying that number with whole numbers. For example, the multiples of 7 include 7, 14, 21, 28, 35, and so forth (multiplying 7 by 1, 2, 3, etc.).
Understanding multiples is extremely important when exploring factors:
- If 7 is a factor of 35, then 35 is a multiple of 7.
- This can be checked by dividing 35 by 7, resulting in 5 with no remainder.
Multiples are often used when calculating common factors or common multiples between sets of numbers. The relationship between factors and multiples is fundamental in topics like factors of a number, least common multiples (LCM), and greatest common divisors (GCD).