Tom has three pieces of rope with lengths of 140 cm, 168 cm and 210 cm. He wishes to cut the three pieces of rope into smaller pieces of equal length with no remainders.

a) what is the greatest possible length of each of the smaller pieces of rope?

b) how many of the smaller pieces of rope of equal lenght can he get?

140 = 2^2 * 5 * 7

168 = 2^3 * 3 * 7
210 = 2 * 3 * 5 * 7

So, what is the biggest number that divides all of those?

14

To find the greatest possible length of each of the smaller pieces of rope, we need to find the greatest common divisor (GCD) of the three given lengths.

a) To find the GCD, we can use the Euclidean algorithm:

Step 1: Find the GCD of 140 cm and 168 cm.
140 cm = 1 * 168 cm + 32 cm

Step 2: Find the GCD of 168 cm and 32 cm.
168 cm = 5 * 32 cm + 8 cm

Step 3: Find the GCD of 32 cm and 8 cm.
32 cm = 4 * 8 cm + 0 cm

As we reach 0 cm, the GCD is the remainder in the previous step, which is 8 cm.

Therefore, the greatest possible length of each of the smaller pieces of rope is 8 cm.

b) To find the number of smaller pieces of rope of equal length that Tom can get, we divide the length of each rope by the greatest common divisor (GCD) found in part (a).

Length of 140 cm rope / GCD of 8 cm = 140 cm / 8 cm = 17.5 pieces (rounded down to the nearest whole number)

Length of 168 cm rope / GCD of 8 cm = 168 cm / 8 cm = 21 pieces

Length of 210 cm rope / GCD of 8 cm = 210 cm / 8 cm = 26.25 pieces (rounded down to the nearest whole number)

Therefore, Tom can get a total of 17 + 21 + 26 = 64 smaller pieces of rope of equal length.

To find the greatest possible length of each of the smaller pieces of rope, we need to find the greatest common divisor (GCD) of the three given lengths (140 cm, 168 cm, and 210 cm).

a) Greatest possible length of each smaller piece: GCD(140, 168, 210)

To find the GCD, we can use the Euclidean algorithm:
Step 1: Find the GCD of 140 and 168.
GCD(140, 168) = 28
Step 2: Find the GCD of 28 and 210.
GCD(28, 210) = 14
Step 3: There is no more division possible, so the GCD is 14.

Therefore, the greatest possible length of each smaller piece of rope is 14 cm.

To find the number of smaller pieces of rope that can be obtained, we need to divide the length of each piece by the greatest common divisor (14 cm).

b) Number of smaller pieces of rope:
- 140 cm : 14 cm = 10 pieces
- 168 cm : 14 cm = 12 pieces
- 210 cm : 14 cm = 15 pieces

Thus, Tom can get 10 pieces of rope each measuring 14 cm from the first piece of rope, 12 pieces of rope each measuring 14 cm from the second piece of rope, and 15 pieces of rope each measuring 14 cm from the third piece of rope.