# Three boys step off together from the same spot. Their steps measure $63 \mathrm{~cm}, 70 \mathrm{~cm}$ and $77 \mathrm{~cm}$ respectively. What is the minimum distance each should cover so that all can cover the distance in complete steps?

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Given:

Three boys step off together from the same spot. Their steps measure $63 \mathrm{~cm}, 70 \mathrm{~cm}$ and $77 \mathrm{~cm}$ respectively.

To do:

We have to find the minimum distance each should cover so that all can cover the distance in complete steps.

Solution:

Required distance will be the LCM of the measure of steps of each boy.

Calculating the LCM of 63, 70 and 77 using prime factorization method:

Writing the numbers as a product of their prime factors:

Prime factorisation of 63:

• $63=3\ \times\ 3\ \times\ 7\ =\ 3^2\ \times\ 7^1$

Prime factorisation of 70:

• $70=2\ \times\ 5\ \times 7=\ 2^1\ \times\ 5^1\ \times\ 7^1$

Prime factorisation of 77:

• $77=7\ \times\ 11\=\ 7^1\ \times\ 11^1$

Multiplying the highest power of each prime number:

• $2^1\ \times\ 3^2\ \times\ 5^1\ \times 7^1\ \times 11^1=\ 6930$

LCM(63, 70, 77)  $=$  6930

Which means required distance $=$ 6930 cm

$=$ 69 m 30 cm (as, 100 cm  $=$  1 m)

So, the minimum distance each should walk so that each can cover the distance in complete steps is 6930 cm or 69 m 30 cm.

Updated on 10-Oct-2022 13:30:51