The question asks: When integer 'i' is divided by 3, the remainder is 1. When 'i' is divided by 4, the remainder is 3. If 'i' is greater than 5 but less than 30, what is its value?

A) 13
B) 15
C) 16
D) 19
E) 23

The correct answer is "D)" and I got this one wrong by answering "A)". I was still unsure after consulting the answer sheet. The solution sheet gives a table with all the possibilities between two equations (3k + 1) and (4m + 3). The table shows different values until there appears to be a match. Additionally, there are separate columns for the values of "3k" and "4m". I don't understand how "3k" starts with a value of 6 and "4m" starts with a value of 4 when the integer 'i' is greater than 5 but less than 30. In short, I am confused as to why the table shows "3k" and "4m" columns and how we obtained 19 as the answer.

In my desperation once again, all I did for this question is to determine the Lowest Common Denominator (LCD) of 3 and 4, and I added 1 (i.e. lowest remainder) to the LCD thereby obtaining 13 as my answer.

The different columns aren't really equations here. (THere's no equal sign, so they can't be.)
Instead we have the integer i divided by 3 with a remainder of 1. Thus, we start with an integer, multiply it by 3 and add 1 to it to arrive at i.
the integer (they're calling it k) multiplied by 3 is 3k. When we add1 to it, we get 3k+1. Those are the first two columns.
They they say that when i is divided by 4, we get a remainder of 3. Same process here. Take an integer, multiply it by 4, and then add 3 to it. Those are the third and fourth column.

THey start with value of k as 2. Multiply it by 3 and we get 6. Add one more and we get 7. That's a possible value for i. (We don't start with k=1 here because we'd be outside the acceptable range of "greater than 5 but less than 30")
Then start with value of m=1, multiply it by 4 and we get 4, and add 3 and we get 7. That's a possible value for i as well.
We need to find a situation where we have a value that works in both situations. 19 shows up both in the 3k+1 column and the 4m+3 column, so it's our answer.

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