Factors Multiples

Factors are numbers that go into another number exactly, leaving no remainder. To find factors, we think of multiplication facts: if a × b = c, both a and b are factors of c. For a child, a simple way is to try small numbers one by one. Start with 1, then 2, then 3, and so on. When division gives a whole number answer, that divisor is a factor. For example, to find factors of 12 try dividing by 1 (12 ÷ 1 = 12), by 2 (12 ÷ 2 = 6), by 3 (12 ÷ 3 = 4), by 4 (12 ÷ 4 = 3), and so on. We stop when we reach a number already seen in a pair because factors come in pairs: (1,12), (2,6), (3,4).

Writing factors in order helps: start with 1, then increasing numbers. Using multiplication tables speeds this up because you spot pairs quickly. For larger numbers, the idea is the same: test small divisors first. A careful check shows that every number has 1 and itself as factors; this is why 1 is always included. Finding factors is useful for sharing objects equally, arranging items in rows and columns, and solving many simple problems in class. Practise finding factors for several small numbers until the method becomes easy and fast.

Multiples are numbers you get by multiplying a number by 1, 2, 3 and so on. For young learners it helps to see multiples as steps on a ladder: each step adds the same number. For example, start at 3 and add 3 repeatedly: 3, 6, 9, 12, 15, ... These are multiples of 3. A good way to make multiples is skip-counting — count forward by the number whose multiples you want. For 4, count 4, 8, 12, 16, 20. Notice that every multiple is bigger than the previous one and that multiples continue forever.

Multiples connect to factors: if 3 is a factor of 12, then 12 is a multiple of 3. Multiples help when planning repeating events like bells or time slots and when looking for numbers that two different counts share. Practice writing lists of multiples for different numbers and spotting where two lists have the same number; those are common multiples. Show this on a number line by marking every multiple of the chosen number — the marked points are evenly spaced. It is useful to know first few multiples of numbers up to 10 by heart because they appear often in school problems and make mental calculations faster.

Even and odd numbers are an easy idea students use every day. An even number can be split into two equal whole groups. Examples: 2, 4, 6, 8, 10. An odd number leaves one extra when split into two groups, for example 1, 3, 5, 7, 9. Another simple test is to look at the last digit: if it is 0, 2, 4, 6 or 8, the number is even; if it is 1, 3, 5, 7 or 9, it is odd. This quick rule works for any whole number.

Even numbers always have 2 as a factor because they are multiples of 2. That means every even number appears in the list of multiples of 2. Odd numbers do not have 2 as a factor. Use objects like buttons to practise: pair up buttons and see whether one remains. When arranging seats in pairs or sharing items between two children, even or odd tells you if sharing is exact. Understanding even and odd is helpful when studying multiplication tables: multiplying by an even number often gives even results. Teachers can ask students to colour even numbers one colour and odd another on a number line to build quick recognition and pattern skills.

Prime and composite numbers tell us how many factors a number has. A prime number has exactly two factors: 1 and itself. Small primes are 2, 3, 5, 7, 11, 13 and so on. Notice that 2 is the only even prime because every other even number has 2 as a factor and thus has more than two factors. A composite number has more than two factors. For example, 6 has factors 1, 2, 3, and 6, so it is composite. The number 1 is special: it has only one factor (1) so it is neither prime nor composite.

To check if a small number is prime, try dividing by 2, 3, 5, and maybe 7 until you either find a divisor or show none work. For students, practise by making a chart from 1 to 50 and marking primes versus composites. Another helpful method is factor checking: try to make arrays other than 1×n. If only 1×n is possible, the number is prime. Recognising primes is useful for later topics and strengthens number sense. Use games and prime hunts to make learning engaging: find as many primes as possible in a range, and explain why each one is prime by showing factor checks.

Factor pairs are two numbers that multiply to give the original number. For example, 18 has pairs (1,18), (2,9) and (3,6). Showing these pairs on paper is good, but making arrays with small objects makes the idea clearer. An array is a rectangle made of dots, blocks or counters arranged in rows and columns. For 12 you can make a 3×4 array by placing 3 rows of 4 dots each. This picture shows both factors 3 and 4 at once.

To find all factor pairs, try arrays with different numbers of rows until no new arrangement is possible. For example, for 20 try 1×20, 2×10, 4×5. If you cannot make any other neat rectangle, you have all pairs. Arrays help children see multiplication and division visually and check work without doing long division. They also make it easy to understand why pairs repeat after the square root: once you reach matching rows and columns reversed, further attempts repeat earlier pairs. Encourage students to use counters to build arrays for several numbers and list the factor pairs they discover.

When two numbers repeat at their own steps, sometimes they land on the same number. Such numbers are common multiples. For instance, multiples of 2 are 2, 4, 6, 8, 10, ... and multiples of 3 are 3, 6, 9, 12, ... The numbers they both have are common multiples; here 6, 12, 18, ... The Least Common Multiple (LCM) is the smallest number, greater than zero, that appears in both lists. LCM is a useful idea when you want two repeating events to meet — for example, two bells that ring after fixed minutes.

To find an LCM for small numbers, make lists of multiples for each number until you see the smallest match. For 4 and 6 list 4, 8, 12 and 6, 12 — the first match is 12. Another way is to use factor knowledge (shared factors) but for Class 4 the listing method works well and builds understanding. Practise with many pairs so students see patterns: if one number divides the other, the larger is the LCM (e.g., LCM of 3 and 9 is 9). Use number lines and charts to show how multiples align and where common multiples occur. This visual work makes LCM easy to spot and remember.

Many story problems use factors and multiples without saying the words. If a problem asks to share things equally, we use factors: how many in each group? If it asks when two repeating events happen together, we use multiples. The steps to solve are simple: read the question carefully, decide if you need factors (equal groups) or multiples (repeating times), make a list or draw arrays, and then write the answer with units (sweets, minutes, rows).

For example, to share 24 apples among 4 children, divide 24 by 4 (or use the factor 4) to find 6 each. For events, if one repeats every 4 minutes and the other every 6 minutes, list multiples to find when they coincide. Encourage drawing pictures: counters for sharing or timelines for repeats. Teach students to check answers by reversing operations — multiply the number per group to see if it gives the total, or test the common time by dividing it by each interval. Word problems build reasoning and show real uses of factors and multiples, so practise a mix of short, picture-based problems and small written ones to build confidence.

Some simple rules make finding factors and multiples faster. These are easy tests teachers give so children can check numbers quickly. For divisibility by 2, look at the last digit: 0, 2, 4, 6 or 8 means divisible by 2. For divisibility by 5, if the last digit is 0 or 5 it is divisible by 5. For 10 the number must end in 0. These rules mean you do not need long division for many checks. Another useful trick is to use multiplication tables and skip-counting to identify multiples quickly; practise counting by 2s, 3s, 4s and 5s until it is automatic.

When looking for factors of a small number, test divisors up to the square root: this means you only need to try small numbers until pairs repeat. For primary classes, focus on the easy endings and using tables. Teachers can show patterns — for example, every second number is even, and numbers ending with 5 always have 5 as a factor. Use short timed games where students test divisibility or list multiples to build speed. These tricks save time in exams and help with mental maths in daily life.