Directions: What is the smallest number, greater than zero, that is divisible by 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10?

### Hint

### Hint

Does the number have to be even or odd? How do you know?

What digit must be in the ones place? How do you know?

The sum of the digits must be equal to a multiple of what number?

What digit must be in the ones place? How do you know?

The sum of the digits must be equal to a multiple of what number?

### Answer

### Answer

2,520

Source: Brian Lack

Rather than being divisible by various numbers, what if there is a remainder?

For example:

Suppose you divide a number by 2 and the remainder is 1.

Suppose, further, that when you divide this number by 3, 4, 5, 6, and 7, the remainder in each case is 1.

What is the smallest positive number that satisfies the constraints above?

Rather than being divisible by various numbers,what if there is a remainder?

For example:

Suppose you divide a number by 2 and the remainder is q.

Suppose,further,that when you divide this number by 3, 4, 5,6, and 7, the remainder in each case is 1.

What is the smallest positive number that satisfies the constraints above?

as the problem is stated, i don’t think there is an answer.

.001 is divisible by all the numbers 1, through 10, as is .0001, and .00001.

it is not *wholly* divisible by any of the numbers; however, this is not a condition in the problem setup.

What about asking, “What is the smallest natural number that yields remainder 1 under division by any set of consecutive natural numbers beginning with 1”?

Now that’s a problem, and was a big hit with my sophomores. Generalization to the extreme.

And then what if we open this up to division by just any set of consecutive natural numbers — not necessarily beginning with 1?