Factors of 420
The factors of 420 are whole numbers that divide it exactly without leaving a remainder. They appear in positive and negative pairs, like (1, 420) or (-1, -420), and are always integers, never fractions or decimals. You can find them using methods such as division or prime factorization. Learning about factors helps build a foundation for understanding divisibility, multiples, and prime numbers. In this guide, we'll explore the different types of factors of 420, including all positive factors, factor pairs, and the prime factorization of 420 with step-by-step explanations and examples.
What are the Factors of 420?
There are 24 factors of 420. The factors of 420 are 1, 2, 3, 4, 5, 6, 7, 10, 12, 14, 15, 20, 21, 28, 30, 35, 42, 60, 70, 84, 105, 140, 210 and 420. Factors can be negative. The negative factors are -1, -2, -3, -4, -5, -6, -7, -10, -12, -14, -15, -20, -21, -28, -30, -35, -42, -60, -70, -84, -105, -140, -210 and -420. All of these numbers divides 420 completely. 420 is a composite number because it has other factors besides 1 and 420.
Factors of 420: 1, 2, 3, 4, 5, 6, 7, 10, 12, 14, 15, 20, 21, 28, 30, 35, 42, 60, 70, 84, 105, 140, 210 and 420
Factor Pairs of 420
The factor pairs of 420 are the pairs of integers that multiply together to give 420. These include both positive and negative combinations. The positive factor pairs of 420 are (1, 420), (2, 210), (3, 140), (4, 105), (5, 84), (6, 70), (7, 60), (10, 42), (12, 35), (14, 30), (15, 28), (20, 21), and the negative factor pairs are (-1, -420), (-2, -210), (-3, -140), (-4, -105), (-5, -84), (-6, -70), (-7, -60), (-10, -42), (-12, -35), (-14, -30), (-15, -28), (-20, -21). Knowing the factor pairs of 420 is useful for learning multiplication, division, and understanding concepts such as the greatest common factor and prime factorization.
Positive Factor Pairs of 420:
| Factor 1 | Factor 2 |
|---|---|
| 1 | 420 |
| 2 | 210 |
| 3 | 140 |
| 4 | 105 |
| 5 | 84 |
| 6 | 70 |
| 7 | 60 |
| 10 | 42 |
| 12 | 35 |
| 14 | 30 |
| 15 | 28 |
| 20 | 21 |
Negative Factor Pairs of 420:
| Factor 1 | Factor 2 |
|---|---|
| -1 | -420 |
| -2 | -210 |
| -3 | -140 |
| -4 | -105 |
| -5 | -84 |
| -6 | -70 |
| -7 | -60 |
| -10 | -42 |
| -12 | -35 |
| -14 | -30 |
| -15 | -28 |
| -20 | -21 |
Prime Factorization of 420
The prime factorization of 420 involves breaking it down into the product of prime numbers. Using division, we find that the prime factors of 420 are 2, 2, 3, 5, 7. Therefore, the prime factorization of 420 is 2^2 × 3 × 5 × 7. Understanding prime factorization helps in finding GCF, LCM, and simplifying fractions.
Prime factors of 420:
2, 2, 3, 5, 7
Prime factorization of 420:
2 × 2 × 3 × 5 × 7
Compact form:
22 × 3 × 5 × 7
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How to Find the Factors of 420?
To find the factors of 420 using the division method efficiently, you only need to check numbers up to the square root of 420. For every number that divides 420 evenly, both it and its corresponding pair (420 ÷ that number) are factors.
Optimized steps to find factors of 420:
- •420 ÷ 1 = 420 → ✅ Factor Pair: (1, 420)
- •420 ÷ 2 = 210 → ✅ Factor Pair: (2, 210)
- •420 ÷ 3 = 140 → ✅ Factor Pair: (3, 140)
- •420 ÷ 4 = 105 → ✅ Factor Pair: (4, 105)
- •420 ÷ 5 = 84 → ✅ Factor Pair: (5, 84)
- •420 ÷ 6 = 70 → ✅ Factor Pair: (6, 70)
- •420 ÷ 7 = 60 → ✅ Factor Pair: (7, 60)
- •420 ÷ 10 = 42 → ✅ Factor Pair: (10, 42)
- •420 ÷ 12 = 35 → ✅ Factor Pair: (12, 35)
- •420 ÷ 14 = 30 → ✅ Factor Pair: (14, 30)
- •420 ÷ 15 = 28 → ✅ Factor Pair: (15, 28)
- •420 ÷ 20 = 21 → ✅ Factor Pair: (20, 21)
This method avoids unnecessary checks and gives all factor pairs quickly. It's especially useful for larger numbers.
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Frequently Asked Questions about factors of 420
What are the factors of 420?
The factors of 420 are 1, 2, 3, 4, 5, 6, 7, 10, 12, 14, 15, 20, 21, 28, 30, 35, 42, 60, 70, 84, 105, 140, 210, 420.
What is the prime factorization of 420?
The prime factorization of 420 is 2 × 2 × 3 × 5 × 7.
How do I find the factors of 420?
To find the factors of 420, start by dividing 420 by every number from 1 up to the square root of 420.
What are factor pairs of 420?
The factor pairs of 420 are (1, 420), (-1, -420), (2, 210), (-2, -210), (3, 140), (-3, -140), (4, 105), (-4, -105), (5, 84), (-5, -84), (6, 70), (-6, -70), (7, 60), (-7, -60), (10, 42), (-10, -42), (12, 35), (-12, -35), (14, 30), (-14, -30), (15, 28), (-15, -28), (20, 21), (-20, -21).
How can I use the factors of 420?
The factors of 420 can be used to simplify fractions, find the greatest common divisor (GCD), and determine multiples.
Are the factors of 420 always positive?
Factors can be both positive and negative. For example, the negative factors of 420 are -1, -2, -3, -4, -5, -6, -7, -10, -12, -14, -15, -20, -21, -28, -30, -35, -42, -60, -70, -84, -105, -140, -210, -420.