Solve, if possible, the following problems. You can use the substitution method,

Question

Solve, if possible, the following problems. You can use the substitution method, or the Chinese Remainder Theorem, or both combined.

(a) 17 thieves divided stolen golden coins into equal piles for each, and there were 3 coins left. While fighting for those 3 coins one thief was killed. The 16 remaining thieves decided to start the process again, and they divided all coins into equal piles for each, but there were 10 coins left. They fought for those 10 coins and one more thief got killed. The 15 remaining thieves decided to divide everything once again, and after placing all the coins into equal piles for each, this time there were no coins left-over, and so they stop killing each other :).

(b) Find the smallest amount of eggs that can be in a basket if: When the eggs are removed 2,3,4,5, and 6 at a time, there are 1,2,3,4, and 5 eggs respectively, left over, and when the eggs are removed 7 at a time, there are none left over.

What is the least possible amount of golden coins that the thieves stole?

Explanation / Answer

let total coins be x.

So clearly by first condition, x=3(mod 17)

i.e. x= 3 + 17a ...........(i)

x=10mod(16)

i.e. x=10+16b ............(ii)

now by third condition

x= 0mod(15) or x= 15c ...........(iii)

Now by (i) and (ii) equations,

3+17a = 10+!6b

or 17a = 7+ 16b=15c

Now let a= 135, b= 143 and c=153, then above equation is full filled

(we guessed it as if 17a= 15 c that is only possible if it is a multiple of 17x 15 = 255 as 17 is prime)

So clearly total coins required be

x= 3+17(135) = 3+2295= 2298

so total coins be 2298.

Answer

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