Question

Three friends plan to go for a morning walk. They step off together and their steps measures 48 cm, 52 cm and 56 cm respectively. What is the minimum distance each should walk so that each can cover the same distance in complete steps ten times?

Answer 1

Let the step lengths of the three friends be \(s_1 = 48\) cm, \(s_2 = 52\) cm, and \(s_3 = 56\) cm. We need to find the minimum distance that each friend can cover in a whole number of steps. This distance will be the Least Common Multiple (LCM) of their step lengths. First, find the prime factorisation of each step length: \(48 = 2 \times 24 = 2 \times 2 \times 12 = 2 \times 2 \times 2 \times 6 = 2 \times 2 \times 2 \times 2 \times 3 = 2^4 \times 3^1\) \(52 = 2 \times 26 = 2 \times 2 \times 13 = 2^2 \times 13^1\) \(56 = 2 \times 28 = 2 \times 2 \times 14 = 2 \times 2 \times 2 \times 7 = 2^3 \times 7^1\) The LCM is the product of the highest powers of all prime factors that appear in any of the numbers: Highest power of 2: \(2^4\) Highest power of 3: \(3^1\) Highest power of 7: \(7^1\) Highest power of 13: \(13^1\) LCM(48, 52, 56) = \(2^4 \times 3^1 \times 7^1 \times 13^1\) LCM = \(16 \times 3 \times 7 \times 13\) LCM = \(48 \times 7 \times 13\) LCM = \(336 \times 13\) 336 \(\times\) 13 ----- 1008 (\(336 \times 3\)) 3360 (\(336 \times 10\)) ----- 4368 So, the minimum distance they can all cover in complete steps is 4368 cm. The question asks: "What is the minimum distance each should walk so that each can cover the same distance in complete steps ten times?" This phrasing can be interpreted as: what is the minimum common distance (D) that they can all cover in complete steps? The "ten times" part implies that this action of covering distance D is repeated ten times. The question asks for D itself. If the question meant that the number of steps taken by each person to cover the common distance must be a multiple of 10, the problem would be different (LCM of 10*48, 10*52, 10*56). However, the phrasing "cover the same distance ... ten times" suggests the "same distance" is the LCM, and this is done 10 times. The "minimum distance" refers to this "same distance". Therefore, the minimum distance is 4368 cm. Number of steps for each friend to cover this distance: Friend 1: \(4368 / 48 = 91\) steps. Friend 2: \(4368 / 52 = 84\) steps. Friend 3: \(4368 / 56 = 78\) steps. Each covers 4368 cm in complete steps. They can do this ten times. The minimum such distance is 4368 cm. \[ \boxed{4368 \text{ cm}} \]