Are the following pairs of numbers co-prime? Guess first and then use prime factorisation to verify your answer.

a. 30 and 45

b. 57 and 85

c. 121 and 1331

d. 343 and 216

Question

Are the following pairs of numbers co-prime? Guess first and then use prime factorisation to verify your answer. a. 30 and 45 b. 57 and 85 c. 121 and 1331 d. 343 and 216

Homi · Accepted Answer

We will find the prime numbers that multiply to make each number. Then we check if they share any prime numbers.

Step 1 — 30 and 45

Let us look at the numbers 30 and 45. Both numbers end in 0 or 5. So, 5 divides both numbers. Also, the sum of digits for 30 is 3. The sum of digits for 45 is 9. Both 3 and 9 can be divided by 3. So, 3 divides both numbers. Since they share 3 and 5 as factors, we guess they are not co-prime. Now we find the prime numbers for 30.

$30 = 2 \times 15$

$= 2 \times 3 \times 5$

$\boxed{\text{Prime factors of } 30 = 2, 3, 5}$

Next, we find the prime numbers for 45.

$45 = 3 \times 15$

$= 3 \times 3 \times 5$

$\boxed{\text{Prime factors of } 45 = 3, 3, 5}$

Both numbers share the prime numbers 3 and 5. So, 30 and 45 are not co-prime.

Step 2 — 57 and 85

Let us look at the numbers 57 and 85. 57 is an odd number. It does not end in 0 or 5. The sum of digits for 57 is $5+7=12$ . 12 can be divided by 3. So, 57 can be divided by 3. $57 = 3 \times 19$ . 85 ends in 5. So, 85 can be divided by 5. $85 = 5 \times 17$ . The prime numbers 3, 19, 5, 17 are all different. We guess they are co-prime. Now we find the prime numbers for 57.

$57 = 3 \times 19$

$\boxed{\text{Prime factors of } 57 = 3, 19}$

Next, we find the prime numbers for 85.

$85 = 5 \times 17$

$\boxed{\text{Prime factors of } 85 = 5, 17}$

The numbers 57 and 85 do not share any prime numbers. So, 57 and 85 are co-prime.

Step 3 — 121 and 1331

Let us look at the numbers 121 and 1331. We know that $11 \times 11 = 121$ . Let us try dividing 1331 by 11. $1331 \div 11 = 121$ . So, both numbers can be divided by 11. We guess they are not co-prime. Now we find the prime numbers for 121.

$121 = 11 \times 11$

$\boxed{\text{Prime factors of } 121 = 11, 11}$

Next, we find the prime numbers for 1331.

$1331 = 11 \times 121$

$= 11 \times 11 \times 11$

$\boxed{\text{Prime factors of } 1331 = 11, 11, 11}$

Both numbers share the prime number 11. So, 121 and 1331 are not co-prime.

Step 4 — 343 and 216

Let us look at the numbers 343 and 216. 343 is an odd number. It does not end in 0 or 5. The sum of digits for 343 is $3+4+3=10$ . 10 cannot be divided by 3. Let us try dividing 343 by 7. $343 = 7 \times 49$ . $49 = 7 \times 7$ . So, $343 = 7 \times 7 \times 7$ . 216 is an even number. It ends in 6. So, 216 can be divided by 2. The sum of digits for 216 is $2+1+6=9$ . 9 can be divided by 3. So, 216 can be divided by 3. The prime numbers for 343 are only 7. The prime numbers for 216 will be 2s and 3s. We guess they are co-prime. Now we find the prime numbers for 343.

$343 = 7 \times 49$

$= 7 \times 7 \times 7$

$\boxed{\text{Prime factors of } 343 = 7, 7, 7}$

Next, we find the prime numbers for 216.

$216 = 2 \times 108$

$= 2 \times 2 \times 54$

$= 2 \times 2 \times 2 \times 27$

$= 2 \times 2 \times 2 \times 3 \times 9$

$= 2 \times 2 \times 2 \times 3 \times 3 \times 3$

$\boxed{\text{Prime factors of } 216 = 2, 2, 2, 3, 3, 3}$

The numbers 343 and 216 do not share any prime numbers. So, 343 and 216 are co-prime.

Answer

(a) Not co-prime. (b) Co-prime. (c) Not co-prime. (d) Co-prime.

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