Question

A number which when divided by $10$ leaves a remainder of $9,$ when divided by $9$ leaves a remainder of $8,$ by $8$ leaves a remainder of $7,$ etc., down to where, when divided by $2,$ it leaves a remainder of $1,$ is:

$59$

$419$

$1259$

$2519$

None of these answers

Answer: D

Concepts:least common multiple modular arithmetic

Difficulty rating: 1580

Small Hint:

Adding $1$ to the desired number removes every listed remainder

Big Hint:

Find $\operatorname{lcm}(2,3,\ldots,10)$

Solution:

If the number is $N,$ then $N+1$ is divisible by every integer from $2$ through $10.$ Their least common multiple is $2^3\cdot3^2\cdot5\cdot7=2520.$ Thus the least positive number fitting all the conditions is $N=2520-1=2519.$

Thus, the correct answer is D.

1951 AMC 12 Problem 37

Answer: D

Difficulty rating: 1580

Small Hint:

Big Hint:

Solution:

Problem 37 in Other Years