Is the first number divisible by the second? Use prime factorisation.

a. 225 and 27

b. 96 and 24

c. 343 and 17

d. 999 and 99

Question

Is the first number divisible by the second? Use prime factorisation. a. 225 and 27 b. 96 and 24 c. 343 and 17 d. 999 and 99

Homi · Accepted Answer

We use prime numbers to check divisibility.

Step 1 — Prime numbers for 225 and 27

Let us find the prime numbers that make up 225. We divide 225 by small prime numbers.

$225 \div 3 = 75$

$75 \div 3 = 25$

$25 \div 5 = 5$

$5 \div 5 = 1$

$\boxed{225 = 3 \times 3 \times 5 \times 5}$

Now, let us find the prime numbers for 27. We divide 27 by small prime numbers.

$27 \div 3 = 9$

$9 \div 3 = 3$

$3 \div 3 = 1$

$\boxed{27 = 3 \times 3 \times 3}$

Step 2 — Compare prime numbers for 225 and 27

We look at the prime numbers for 27. It has three 3s. We look at the prime numbers for 225. It has two 3s. To be divisible, 225 needs three 3s. It only has two 3s. So, 225 is not divisible by 27.

Step 3 — Prime numbers for 96 and 24

Let us find the prime numbers that make up 96. We divide 96 by small prime numbers.

$96 \div 2 = 48$

$48 \div 2 = 24$

$24 \div 2 = 12$

$12 \div 2 = 6$

$6 \div 2 = 3$

$3 \div 3 = 1$

$\boxed{96 = 2 \times 2 \times 2 \times 2 \times 2 \times 3}$

Now, let us find the prime numbers for 24. We divide 24 by small prime numbers.

$24 \div 2 = 12$

$12 \div 2 = 6$

$6 \div 2 = 3$

$3 \div 3 = 1$

$\boxed{24 = 2 \times 2 \times 2 \times 3}$

Step 4 — Compare prime numbers for 96 and 24

We look at the prime numbers for 24. It has three 2s and one 3. We look at the prime numbers for 96. It has five 2s and one 3. All prime numbers of 24 are in 96. So, 96 is divisible by 24.

Step 5 — Prime numbers for 343 and 17

Let us find the prime numbers that make up 343. We divide 343 by small prime numbers.

$343 \div 7 = 49$

$49 \div 7 = 7$

$7 \div 7 = 1$

$\boxed{343 = 7 \times 7 \times 7}$

Now, let us find the prime numbers for 17. The number 17 is a prime number itself.

$\boxed{17 = 17}$

Step 6 — Compare prime numbers for 343 and 17

We look at the prime numbers for 17. It has one 17. We look at the prime numbers for 343. It has three 7s. The prime number 17 is not in 343. So, 343 is not divisible by 17.

Step 7 — Prime numbers for 999 and 99

Let us find the prime numbers that make up 999. We divide 999 by small prime numbers.

$999 \div 3 = 333$

$333 \div 3 = 111$

$111 \div 3 = 37$

$37 \div 37 = 1$

$\boxed{999 = 3 \times 3 \times 3 \times 37}$

Now, let us find the prime numbers for 99. We divide 99 by small prime numbers.

$99 \div 3 = 33$

$33 \div 3 = 11$

$11 \div 11 = 1$

$\boxed{99 = 3 \times 3 \times 11}$

Step 8 — Compare prime numbers for 999 and 99

We look at the prime numbers for 99. It has two 3s and one 11. We look at the prime numbers for 999. It has three 3s and one 37. The prime number 11 is not in 999. So, 999 is not divisible by 99.

Answer

(a) 225 is not divisible by 27. (b) 96 is divisible by 24. (c) 343 is not divisible by 17. (d) 999 is not divisible by 99.

Question 33

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