Kenji has a box containing many blue rods, each one 252 cm long. Mei-Ling has a box containing many green rods, each one 408 cm long. They each start making a long line of rods of their colour by laying rods end to end. After a while, their lines are exactly the same length. What is this length, if Kenji and Mei-Ling have each used the minimum number of rods? How many rids did each use? Five a complete, well-organized solution.
you didnt make roblox, people always try to take credit these days,
Isn't 51 divisible by 3 and 17?
you are looking for the least common multiple of 252 and 408
The way I do such problems is to factor each
252 = 2*126
= 2*2*63
= 2*2*3*21
= 2*2*3*3*7
then
408 = 2*204
= 2*2*102
= 2*2*2*51
so we need
2*2*2 * 3*3 * 7 * 51 = 8*9*7*51=25704
check
25704 / 252 = 102 blue
25704 / 408 = 63 green
mmmmmmmmmmmmmmmmmmmmm
Is there anything else I can help you with?
with correction (question says minimum rods)
252 = 2*126
= 2*2*63
= 2*2*3*21
= 2*2*3*3*7
then
408 = 2*204
= 2*2*102
= 2*2*2*51
= 2*2*2*17*3
so we need
2*2*2 * 3*3 * 7 * 17 = 8*9*7*17 = 8568
check
8568 / 252 = 34 blue
8568 / 408 = 21 green
You are absolutely correct! Thank you for pointing out my mistake. The least common multiple of 252 and 408 is indeed 8568, and the minimum number of rods needed by Kenji and Mei-Ling are 34 and 21 respectively.
To solve this problem, we need to find the least common multiple (LCM) of the lengths of the blue and green rods.
The LCM is the smallest number that both lengths can divide evenly into.
Step 1: Find the prime factors of each rod length:
- The prime factors of 252 are 2, 2, 3, 3, and 7.
- The prime factors of 408 are 2, 2, 2, 3, 17.
Step 2: Identify the shared and unshared prime factors.
- The shared prime factors are 2 and 3.
- The unshared prime factors for 252 are 2 and 7.
- The unshared prime factors for 408 are 2 and 17.
Step 3: Multiply the shared and unshared prime factors to find the LCM.
- The shared prime factors are 2 and 3, so we multiply them together: 2 * 3 = 6.
- For the unshared prime factors, we take the largest power of each prime:
- For 252, we have 2^2 and 7^1.
- For 408, we have 2^3 and 17^1.
- Multiply the unshared prime factors together: 2^3 * 7 * 17 = 952.
Step 4: Multiply the LCM by the number of rods needed for each color.
- For the blue rods, multiply the LCM by the number of blue rods equivalent to the LCM: 952 * 1 = 952.
- For the green rods, multiply the LCM by the number of green rods equivalent to the LCM: 952 * 2 = 1904.
Step 5: Calculate the total length of rods used by each person.
- Kenji used 952 cm of blue rods.
- Mei-Ling used 1904 cm of green rods.
Step 6: Find the length of the equal line of rods.
- Since both Kenji and Mei-Ling used the minimum number of rods, the lengths of their lines will be equal.
- The total length of the equal line of rods is 952 cm.
To summarize:
- Kenji used 952 cm of blue rods.
- Mei-Ling used 1904 cm of green rods.
- The length of the equal line of rods is 952 cm.