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.)

For any two numbers, $\text{HCF} \times \text{LCM}$ equals the product of the numbers. So the product is $6 \times 72 = 432$, and the other number is $432 \div 24 = 18$. (Check: $\text{HCF}(24, 18) = 6$ and $\text{LCM}(24, 18) = 72$.)

The smallest number divisible by all three is their LCM. Using primes, $4 = 2^2$, $6 = 2\times 3$, $9 = 3^2$, so the LCM is $2^2\times 3^2 = 36$. (12 is divisible by 4 and 6 but not by 9.)

The side length is the square root of the area: $\sqrt{100} = 10$ m. A square has 4 equal sides, so the perimeter is $4 \times 10 = 40$ m. (10 m is just one side; using the area 100 as a length leads to the wrong 400.)

List the factors in pairs: $1\times 36$, $2\times 18$, $3\times 12$, $4\times 9$, $6\times 6$. Since $6\times 6$ repeats the 6, there are 9 different factors. (A perfect square always has an odd number of factors.)

Check numbers in order and count their factors. 16 is the first with exactly five: $1, 2, 4, 8, 16$. (12 has six factors and $32 = 2^5$ has six; only perfect squares can have an odd number of factors.)

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

The HCF of 18 and 24 is 6. Divide the top and bottom by 6: $\dfrac{18 \div 6}{24 \div 6} = \dfrac{3}{4}$. ($\dfrac{9}{12}$ and $\dfrac{6}{8}$ are only partly simplified.)

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 when its digit sum is. Here the digit sum is $4 + \square + 2 = 6 + \square$, which is a multiple of 3 when $\square = 0, 3, 6$ or $9$. That is 4 possible digits.

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.)

A number has exactly three factors only when it is the square of a prime (its factors are 1, the prime, and the square). Below 30 these are $4 = 2^2$, $9 = 3^2$ and $25 = 5^2$, so there are 3. ($49 = 7^2$ is already above 30.)

The tile side must divide both 30 and 18 exactly, so it is their highest common factor. $30 = 2\times 3\times 5$ and $18 = 2\times 3^2$, giving HCF $= 2\times 3 = 6$ cm. (A 3 cm tile fits too, but 6 cm is the largest.)

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.

The lowest common denominator is $\text{LCM}(6, 9) = 18$. Rewrite: $\dfrac{1}{6} = \dfrac{3}{18}$ and $\dfrac{1}{9} = \dfrac{2}{18}$. Add: $\dfrac{3}{18} + \dfrac{2}{18} = \dfrac{5}{18}$. (Adding the numerators and denominators to get $\dfrac{2}{15}$ is the common error.)

The perfect squares just above 100 are $11^2 = 121$, $12^2 = 144$, $13^2 = 169$ and $14^2 = 196$. The next one, $15^2 = 225$, is above 200, so there are 4.

Both numbers must be prime and add to 24. 11 and 13 are both prime and $11 + 13 = 24$. (In every other pair at least one number is composite: 8, 9, 10, 12, 14, 15 and 16 are all composite.)

They ring together at common multiples of 6 and 8. The lowest common multiple of 6 and 8 is 24, so they next ring together after 24 minutes. (14 comes from adding 6 and 8 instead of finding the LCM.)

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.