Can someone please explain how to solve the following question?
"How many integers between 26 and 92, inclusive, have a remainder of 6 when divided by 13? Name those integers."
Thank you in advance
To solve this question, you need to find the integers between 26 and 92 (inclusive) that have a remainder of 6 when divided by 13. Here's how you can do it:
1. Start by finding the first multiple of 13 that is greater than or equal to 26. Divide 26 by 13: 26 ÷ 13 = 2 with a remainder of 0. Since the remainder is 0, 26 itself is divisible by 13. So, the first multiple of 13 that is greater than or equal to 26 is 26.
2. Similarly, find the last multiple of 13 that is less than or equal to 92. Divide 92 by 13: 92 ÷ 13 = 7 with a remainder of 1. Since the remainder is not 0, 92 is not divisible by 13. So, the last multiple of 13 that is less than or equal to 92 is 91.
3. Now, we have the range of multiples of 13 between 26 and 92: {26, 39, 52, 65, 78, 91}.
4. Since we are looking for integers that have a remainder of 6 when divided by 13, we need to eliminate the multiples of 13 that have different remainders. Out of the six numbers we found, only three have a remainder of 6 when divided by 13: 39, 65, and 91.
Therefore, there are three integers between 26 and 92 (inclusive) that have a remainder of 6 when divided by 13, and they are: 39, 65, and 91.
To solve this question, we need to find the integers between 26 and 92 (inclusive) that have a remainder of 6 when divided by 13.
To do this, we can use the concept of modular arithmetic. Modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after a certain value called the modulus.
In this case, the modulus is 13 because we are dividing by 13. So, we are looking for integers that, when divided by 13, leave a remainder of 6.
To find the integers, we can start with the lower bound (26) and check each integer up to the upper bound (92). For each integer, we divide it by 13 and check if the remainder is 6.
Let's go step by step:
1. Start with the lower bound, which is 26.
2. Divide 26 by 13: 26 ÷ 13 = 2 with no remainder.
Since the remainder is not 6, move on to the next integer.
3. Next is 27.
4. Divide 27 by 13: 27 ÷ 13 = 2 with a remainder of 1.
Again, the remainder is not 6, so we move on to the next integer.
5. Continue this process for all values between 26 and 92, checking if each integer divided by 13 has a remainder of 6.
Once we reach the upper bound (92), check the remainder for that integer.
92 ÷ 13 = 7 with a remainder of 11.
Since the remainder is not 6, we have checked all the integers between 26 and 92 and none have a remainder of 6 when divided by 13.
Therefore, there are no integers between 26 and 92 (inclusive) that satisfy the condition of having a remainder of 6 when divided by 13.
32/13 = 2 remainder 6
32 + 13 = 45
45/13 = 3 remainder 6
Take it from there.