Let $x$ be the least number which when divided by $10,12,14$ and $16$ leaves the remainders $2,4,6$ and $8$, respectively, but $x$ is divisible by $17$. When $x$ is divided by $52$, the quotient is:

Notice the constant difference between divisor and remainder:
$$10-2=8$$
$$12-4=8$$
$$14-6=8$$
$$16-8=8$$
Thus, the general form of the number is:
$$x=\text{LCM}(10,12,14,16)\cdot k-8$$
Let's compute the LCM:$10=2\times 5$$12=2^2\times 3$$14=2\times 7$$16=2^4$ $
$\text{LCM}=2^4\times 3\times 5\times 7=16\times 3\times 5\times 7=1680$$So,$ x = 1680k - 8$.Wearegiventhat$ x$isexactlydivisibleby$ 17$:$$\frac{1680k-8}{17}$$Dividing$ 1680$by$ 17$:$$1680=17\times 98+14$$Sotheremainderexpressionsimplifiesto:$$\frac{14k-8}{17}$$Le{t}^{\prime }stestintegervaluesof$ k$tofindwhen$ (14k - 8)$isdivisibleby$ 17$:For$ k = 1: 14(1) - 8 = 6$For$ k = 2: 14(2) - 8 = 20$For$ k = 3: 14(3) - 8 = 34$,whichisexactlydivisibleby$ 17$($ 34 / 17 = 2$).Thus,thelowestvalueisat$ k = 3$:$$x=1680(3)-8=5040-8=5032$$Now,wefindthequotientwhen$ x$isdividedby$ 52$:$$\text{Quotient}=\lfloor \frac{5032}{52}\rfloor =96$$Le{t}^{\prime }sverify:$ 52 \times 96 = 4992$.Remainderis$ 5032 - 4992 = 40$.Sothequotientis$ 96$.