What is the Smallest Number That Has a Remainder of 2 When Divided by 34 and 15?

The problem at hand involves finding the smallest number that leaves a remainder of 2 when divided by 34 and 15. This requires an understanding of least common multiples (LCMs) and some strategic thinking. Let's break down the solution step-by-step.

Understanding the Problem and the Least Common Multiple (LCM)

The first step is to find the LCM of the given numbers, which are 34 and 15. The LCM is the smallest number that is divisible by both 34 and 15. Let's calculate the LCM of 34 and 15.

Calculating the LCM of 34 and 15

First, factorize the numbers:

The LCM is the product of the highest powers of all prime factors involved:

LCM 2 x 3 x 5 x 17 510

However, since we are looking for a remainder of 2 when divided by 34 and 15, we need to find a number that is 2 more than a multiple of the LCM 510. But let's simplify our approach by finding the LCM of 34 and 15 directly and then adding 2.

LCM of 34 and 15 510

So, the number we are looking for can be expressed as 510k 2, where k is a non-negative integer.

Finding the Smallest 4-Digit Number

We need a 4-digit number that fits the criteria. Start by finding the smallest 4-digit number, which is 1000. However, 1000 is not a multiple of 510 2.

Next, find the smallest multiple of 510 that is close to 1000:

1000 ÷ 510 ≈ 1.96

The smallest integer greater than 1.96 is 2. Multiply 510 by 2:

2 x 510 1020

Now, add 2 to 1020 to get the desired number:

1020 2 1022

1022 is the smallest 4-digit number that leaves a remainder of 2 when divided by 34 and 15.

Exploring Other Approaches

Consider the possibility of a smaller 4-digit number. Let's look at other methods and confirm our answer:

Multiple of 60

Another approach is to find the LCM of 3, 4, 5, 15, and 2.

LCM(3, 4, 5, 15, 2) 60

Since we need a remainder of 2, the number can be expressed as 60k 2, where k is a non-negative integer.

Using trial and error, the smallest k such that 60k 2 is a 4-digit number is 17:

60 x 17 2 1020 2 1022

So, the number 1022 is confirmed again as the smallest 4-digit number that fits the criteria.

General Solution

In general, the smallest number that leaves a remainder of 2 when divided by any combination of 34 and 15 can be expressed as 60k 2, where k is an integer. The smallest 4-digit number is 1022.

Thus, the answer to the problem is 1022, and we have confirmed this answer through multiple approaches: the LCM method, the direct calculation, and the 60k 2 method.

For those interested, the next few such numbers include 1022, 1182, 1342, and 1502.