Factors, Multiples and Prime Numbers
Factors are numbers that divide exactly into another number without leaving a remainder. Think of them as the building blocks of numbers - like how 2 and 3 are factors of 6 because 2×3=6.
The Highest Common Factor (HCF) is the biggest number that divides into two or more numbers. Multiples are what you get from times tables - so multiples of 5 are 5, 10, 15, 20, etc. The Lowest Common Multiple (LCM) is the smallest number that appears in the times tables of all your given numbers.
Prime numbers only have exactly two factors: themselves and 1. Remember, 1 isn't prime because it only has one factor! The first few primes are 2, 3, 5, 7, 11, 13. When you break numbers down into prime factors, you're finding which primes multiply together to make your original number.
Quick Tip: Always write factors in pairs - it makes finding all of them much easier!
Powers and Indices
Square numbers come from multiplying a number by itself, like 5×5=25. Cube numbers involve multiplying three times: 4×4×4=64. The square root and cube root work backwards - asking "what number gives me this when squared or cubed?"
When working with powers (indices), there are three key rules to master. For multiplication with the same base, you add the powers: a⁵ × a³ = a⁸. For division, you subtract: a⁷ ÷ a² = a⁵. When raising a power to another power, you multiply them: (a³)⁴ = a¹².
Don't confuse x² with 2x! Squaring means multiplying by itself, not by 2. Also remember that any number to the power of 1 equals itself, and any number to the power of 0 equals 1.
Watch Out: p³ when p=2 gives you 2×2×2=8, not 2×3=6!
Fractions Fundamentals
A fraction shows division - the numerator (top number) divided by the denominator (bottom number). Unit fractions have 1 on top, like ½ or ¼.
To simplify fractions, divide both top and bottom by their highest common factor. Equivalent fractions represent the same value: ½ = 2/4 = 10/20. When comparing fractions, convert them to have the same denominator first.
For adding or subtracting fractions, find the LCM of the denominators to create a common denominator. Convert each fraction, then add or subtract only the numerators whilst keeping the denominator the same.
Remember: When rounding money, always show two decimal places - write £27.40, not £27.4!
Rounding and Estimating
Rounding makes numbers easier to work with whilst keeping their value close to the original. If the digit after your rounding position is 5 or more, round up; if it's less than 5, round down.
Significant figures are the meaningful digits in a number. The first significant figure can't be zero, and trailing zeros in decimals don't count. When estimating calculations, round each number to 1 significant figure first.
Decimal places count positions after the decimal point. Always check if you need a specific number of decimal places in your answer, especially with money questions.
Pro Tip: For quick estimates, use the ≈ symbol meaning "approximately equal to" - it shows you're estimating, not calculating exactly!