Question 1

We will find the prime factors of each number. Then we list all factors.

Step 1 — Prime Factorisation of 90

Let us find the prime factors of 90. We divide 90 by the smallest prime numbers.

$90 \div 2 = 45$

$45 \div 3 = 15$

$15 \div 3 = 5$

$5 \div 5 = 1$

So, the prime factorisation of 90 is:

$\boxed{90 = 2 \times 3 \times 3 \times 5}$

Step 2 — Listing Factors of 90

We use the prime factors to find all combinations. The prime factors are 2, 3, and 5. The powers are $2^1$, $3^2$, $5^1$. We can form factors by taking $2^a \times 3^b \times 5^c$. Here $a$ can be 0 or 1. $b$ can be 0, 1, or 2. $c$ can be 0 or 1. Let us list all these combinations in increasing order.

$\boxed{\text{Factors of 90: 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90}}$

Step 3 — Prime Factorisation of 105

Let us find the prime factors of 105. We divide 105 by the smallest prime numbers.

$105 \div 3 = 35$

$35 \div 5 = 7$

$7 \div 7 = 1$

So, the prime factorisation of 105 is:

$\boxed{105 = 3 \times 5 \times 7}$

Step 4 — Listing Factors of 105

We use the prime factors to find all combinations. The prime factors are 3, 5, and 7. The powers are $3^1$, $5^1$, $7^1$. We can form factors by taking $3^a \times 5^b \times 7^c$. Here $a, b, c$ can be 0 or 1. Let us list all these combinations in increasing order.

$\boxed{\text{Factors of 105: 1, 3, 5, 7, 15, 21, 35, 105}}$

Step 5 — Prime Factorisation of 132

Let us find the prime factors of 132. We divide 132 by the smallest prime numbers.

$132 \div 2 = 66$

$66 \div 2 = 33$

$33 \div 3 = 11$

$11 \div 11 = 1$

So, the prime factorisation of 132 is:

$\boxed{132 = 2 \times 2 \times 3 \times 11}$

Step 6 — Listing Factors of 132

We use the prime factors to find all combinations. The prime factors are 2, 3, and 11. The powers are $2^2$, $3^1$, $11^1$. We can form factors by taking $2^a \times 3^b \times 11^c$. Here $a$ can be 0, 1, or 2. $b$ can be 0 or 1. $c$ can be 0 or 1. Let us list all these combinations in increasing order.

$\boxed{\text{Factors of 132: 1, 2, 3, 4, 6, 11, 12, 22, 33, 44, 66, 132}}$

Step 7 — Prime Factorisation of 360

Let us find the prime factors of 360. We divide 360 by the smallest prime numbers.

$360 \div 2 = 180$

$180 \div 2 = 90$

$90 \div 2 = 45$

$45 \div 3 = 15$

$15 \div 3 = 5$

$5 \div 5 = 1$

So, the prime factorisation of 360 is:

$\boxed{360 = 2 \times 2 \times 2 \times 3 \times 3 \times 5}$

Step 8 — Listing Factors of 360

We use the prime factors to find all combinations. The prime factors are 2, 3, and 5. The powers are $2^3$, $3^2$, $5^1$. We can form factors by taking $2^a \times 3^b \times 5^c$. Here $a$ can be 0, 1, 2, or 3. $b$ can be 0, 1, or 2. $c$ can be 0 or 1. Let us list all these combinations in increasing order.

$\boxed{\text{Factors of 360: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, 360}}$

Step 9 — Prime Factorisation of 840

Let us find the prime factors of 840. We divide 840 by the smallest prime numbers.

$840 \div 2 = 420$

$420 \div 2 = 210$

$210 \div 2 = 105$

$105 \div 3 = 35$

$35 \div 5 = 7$

$7 \div 7 = 1$

So, the prime factorisation of 840 is:

$\boxed{840 = 2 \times 2 \times 2 \times 3 \times 5 \times 7}$

Step 10 — Listing Factors of 840

We use the prime factors to find all combinations. The prime factors are 2, 3, 5, and 7. The powers are $2^3$, $3^1$, $5^1$, $7^1$. We can form factors by taking $2^a \times 3^b \times 5^c \times 7^d$. Here $a$ can be 0, 1, 2, or 3. $b, c, d$ can be 0 or 1. Let us list all these combinations in increasing order.

$\boxed{\text{Factors of 840: 1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 15, 20, 21, 24, 28, 30, 35, 40, 42, 56, 60, 70, 84, 105, 120, 140, 168, 210, 280, 420, 840}}$

Answer

(a) Prime factorisation: $90 = 2 \times 3 \times 3 \times 5$. Factors of 90 are 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, and 90. Total factors = 12. (b) Prime factorisation: $105 = 3 \times 5 \times 7$. Factors of 105 are 1, 3, 5, 7, 15, 21, 35, and 105. Total factors = 8. (c) Prime factorisation: $132 = 2 \times 2 \times 3 \times 11$. Factors of 132 are 1, 2, 3, 4, 6, 11, 12, 22, 33, 44, 66, and 132. Total factors = 12. (d) Prime factorisation: $360 = 2 \times 2 \times 2 \times 3 \times 3 \times 5$. Factors of 360 are 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, and 360. Total factors = 24. (e) Prime factorisation: $840 = 2 \times 2 \times 2 \times 3 \times 5 \times 7$. Factors of 840 are 1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 15, 20, 21, 24, 28, 30, 35, 40, 42, 56, 60, 70, 84, 105, 120, 140, 168, 210, 280, 420, and 840. Total factors = 32.