16. What is the largest multiple of 7 less than 10,000 which can be expressed as the sum of squares of three consecutive numbers?
16. 8750
Let the number be expressed as $a^{2}+(a+1)^{2}+(a+2)^{2}$, where $a$ is an integer. It may be checked that this expression is a multiple of 7 if and only if the remainder when a is divided by 7 is 1 or 4 . In the former case, the largest possible value of a that places the value of the expression within bounds is 50 , which gives the value $50^{2}+51^{2}+52^{2}=7805$. In the latter case, the largest such value of $a$ is 53 , which gives the value $53^{2}+54^{2}+55^{2}=8750$.