The three friends take steps of lengths $48$ cm, $52$ cm, and $56$ cm respectively.
To find the minimum distance each should walk so that they cover the same distance in complete steps, we need to find the Least Common Multiple (LCM) of $48$, $52$, and $56$.
First, we find the prime factorization:
$48 = 2^4 \times 3$
$52 = 2^2 \times 13$
$56 = 2^3 \times 7$
The LCM is given by taking the highest powers of all prime factors:
$LCM(48, 52, 56) = 2^4 \times 3 \times 7 \times 13$
Calculating step-by-step:
$2^4 = 16$
$16 \times 3 = 48$
$48 \times 7 = 336$
$336 \times 13 = 4368$
Thus, the minimum distance each should walk is $4368$ cm.
Since they need to cover this distance ten times, the total distance each friend should walk is:
$10 \times 4368 = 43680$ cm