L.C.M. of 3,4,5,6 = 60 Number is of the form = $$60 k_1 + 2$$ -------------(i) When divided by 11, it leaves 0 remainder so number will also be of the form = $$11 k_2$$ ---------(ii) Hence equating (i) and (ii), we get, $$60k_1 + 2 = 11k_2$$ $$60k_1 - 11k_2 = -2$$ or $$11k_2 - 60k_1 = 2$$ -----------(iii) It means $$60k_1$$ will leave remainder 9 when divide by 11. But, in (iii) 60 leaves 5 as remainder when divided by 11 $$ herefore$$ By remainder root $$frac{5 k_1}{11}$$ should leave remainder as 9 or -2 => Possible values of $$K_1 = 4, 15, 26, 37, 48, 59$$ $$ herefore$$ Required value = $$60 imes 59 + 2 = 3540 + 2 = 3542$$