The given problem is regarding solving of linear congruences. Given congruences are as follows

X= 3mod(17); X= 10 mod(16) and X= 0 mod(15)

If k is any integer Z (Set of integers) then X= 3+17k. Equating this to the congruence X=0mod(15), 3+17k= 0mod(15)

17k= -3mod(15)

Add to this 0=105mod(15)

Hence 17k= 102mod(15)

Or, k= 6mod(15)

= 6+15 l , where l Z (Set of integers)

This gives X= 3+ 17( 6 + 15 l)

= 105 +255l

Now setting this to X= 10mod(16), 105+255l = 10 mod(16)

255l= -95mod(16)

Add to this 0= -160mod(16)

255l= -255 mod(16)

Or l = -1 mod(16)

= -1 + 16 m , where m Z (Set of integers)

Now substitute for l in X=105 +255l, X= 105 +255 (-1 + 16m)

= -150 + 4080m,

= 4080m-150

which gives the required solutions for X, where m= 1,2,3 ….etc

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