# Problem of the Day

If for some integer, , which of the following expressions could be equal to ?

A.
B.
C.
D.
E.

[latexpage]

E. $11n+10$

See the Solution

### Solution

[latexpage]

$1154\cdot1156=(1155-1)(1155+1)=1155^2-1$. Thus, this product is one less than a multiple of $1155$.

Also, note that $1155=3\cdot5\cdot7\cdot11$, so the product is also one less than a multiple of 3, 5, 7, and 11. Answer choice E is 10 more than a multiple of 11, which is the same as being one less than a multiple of 11. For example, 21 is one less than a multiple of 11 (22), and ten more than a multiple of 11 (11). All of the other answer choices are not consistent with the fact that the product is one less than a multiple of 3, 5, 7, and 11, so ‘E’ is the answer.