Number Properties

A prime number has exactly two factors: 1 and itself. 13 qualifies. ($9 = 3\times 3$, $15 = 3\times 5$, $21 = 3\times 7$ are composite, and 1 has only one factor so it is not prime.)

Factors divide 12 exactly, in pairs: $1\times 12$, $2\times 6$, $3\times 4$. So the factors are 1, 2, 3, 4, 6, 12. (12, 24, 36 are multiples of 12, not factors.)

Multiples of 7 are 7, 14, 21, 28, 35, 42, ... and $42 = 6\times 7$. (The others are not in the 7 times table.)

$5^2 = 5\times 5 = 25$. (10 comes from $5\times 2$; the index means "how many times to multiply", not "multiply by".)

$\sqrt{49}$ is the number that multiplies by itself to give 49. Since $7\times 7 = 49$, $\sqrt{49} = 7$. (24.5 halves 49; square root is the opposite of squaring.)

Factors of 12: 1, 2, 3, 4, 6, 12. Factors of 18: 1, 2, 3, 6, 9, 18. The largest one they share is 6. (36 is a common multiple, not a common factor.)

Multiples of 4: 4, 8, 12, 16, ... Multiples of 6: 6, 12, 18, ... The smallest they share is 12. (24 is a common multiple but not the lowest.)

A composite number has more than two factors. $15 = 3\times 5$, so it is composite. (11, 13, 17 and 2 are all prime.)

The 2 is repeated three times, so it becomes $2^3$: $2\times 2\times 2\times 5 = 2^3 \times 5$. ($6\times 5$ wrongly adds the three 2s instead of multiplying them.)

$3^3 = 3\times 3\times 3 = 27$. (9 is $3^2$; 6 comes from $3\times 2$.)

By definition, a prime number has exactly two factors: 1 and the number itself (for example the factors of 7 are 1 and 7).

When two numbers share no common factors, their LCM is simply their product: $3\times 5 = 15$. (1 is their HCF, not their LCM.)

The nearest perfect squares are $25 = 5^2$ and $36 = 6^2$. Since $25 < 30 < 36$, $\sqrt{30}$ lies between 5 and 6 (about 5.5).

A number is divisible by 3 if its digits add to a multiple of 3. For 51: $5 + 1 = 6$, which is divisible by 3, so 51 is too ($51 = 3\times 17$).

Break 60 into primes: $60 = 2\times 30 = 2\times 2\times 15 = 2\times 2\times 3\times 5 = 2^2 \times 3 \times 5$. (The other options still contain composite numbers like 10, 15 or 30.)

$24 = 2^3\times 3$ and $36 = 2^2\times 3^2$. The shared factors are $2^2\times 3 = 12$, so the HCF is 12. (72 is the LCM.)

Multiples of 8: 8, 16, 24, 32, ... Multiples of 12: 12, 24, 36, ... The smallest they share is 24. (4 is their HCF; 96 is a common multiple but not the lowest.)

A prime number has exactly two different factors. The only factor of 1 is 1 itself, so it has just one factor and is neither prime nor composite.

A number divisible by both 2 and 3 must be a multiple of their LCM, which is $2\times 3 = 6$. (It need not be divisible by 12 or 9, for example 18 is divisible by 6 but not 12.)

Every whole number above 1 is prime (exactly two factors) or composite (more than two). The traps: 9 is odd but not prime; 2 is even but prime; and 1 is neither prime nor composite.

Common multiples are numbers in both times tables: 12, 24, 36, ... The first two are 12 and 24. (18 is a multiple of 6 but not of 4.)

$12\times 12 = 144$, so $\sqrt{144} = 12$. (72 halves 144; square root undoes squaring.)

The factors of 9 are 1, 3 and 9, which is exactly three. (6 and 8 each have four factors, 12 has six, and 7 is prime with two. A number has an odd number of factors only when it is a perfect square.)

They ring together at common multiples of 4 and 6. The lowest common multiple of 4 and 6 is 12, so they next ring together after 12 minutes.

An HCF of 1 means the only factor the two numbers share is 1, so they are called co-prime (for example 8 and 9). The numbers themselves do not have to be prime.