How many integers bewteen 200 and 500 inclusive leave a remainder 1 when divided by 7 and a remainder 3 when divided by 4?

Find the smallest positive integer that leaves a remainder 5 when divided by 7, a remainder 6 when divided by 11, and a remainder 4 when divided by 13.

To find the number of integers between 200 and 500 (inclusive) that satisfy the given conditions, we need to find the range of numbers that leave a remainder of 1 when divided by 7 and a remainder of 3 when divided by 4.

1) Finding the range of numbers with remainder 1 when divided by 7:
To find the range, we need to determine the smallest and largest numbers within the given range that satisfy the condition.

The smallest number in the given range is 200. To find the smallest number that leaves a remainder of 1 when divided by 7, we can start from 200 and increment by 7 until it satisfies the condition.

200 mod 7 = 2 (does not satisfy)
200 + 7 = 207 mod 7 = 2 (does not satisfy)
200 + 14 = 214 mod 7 = 0 (does not satisfy)
200 + 21 = 221 mod 7 = 6 (does not satisfy)
200 + 28 = 228 mod 7 = 5 (does not satisfy)
200 + 35 = 235 mod 7 = 5 (does not satisfy)
200 + 42 = 242 mod 7 = 4 (does not satisfy)
200 + 49 = 249 mod 7 = 6 (does not satisfy)
200 + 56 = 256 mod 7 = 6 (does not satisfy)
200 + 63 = 263 mod 7 = 5 (does not satisfy)
200 + 70 = 270 mod 7 = 4 (does not satisfy)
200 + 77 = 277 mod 7 = 6 (does not satisfy)
200 + 84 = 284 mod 7 = 5 (does not satisfy)
200 + 91 = 291 mod 7 = 1 (satisfies!)

So, the smallest number in the given range that leaves a remainder of 1 when divided by 7 is 291.

To find the largest number in the given range that satisfies the condition, we can start from 500 and decrement by 7 until it satisfies the condition.

500 mod 7 = 3 (does not satisfy)
500 - 7 = 493 mod 7 = 0 (does not satisfy)
500 - 14 = 486 mod 7 = 6 (does not satisfy)
500 - 21 = 479 mod 7 = 6 (does not satisfy)
500 - 28 = 472 mod 7 = 2 (does not satisfy)
500 - 35 = 465 mod 7 = 5 (does not satisfy)
500 - 42 = 458 mod 7 = 4 (does not satisfy)
500 - 49 = 451 mod 7 = 3 (satisfies!)

So, the largest number in the given range that leaves a remainder of 1 when divided by 7 is 451.

To find the number of integers that satisfy the given condition, we can subtract the smallest number found (291) from the largest number found (451) and then divide the result by 7 since the pattern repeats every 7 numbers.

Number of integers = (451 - 291) / 7 + 1 = 160 / 7 + 1 = 22 + 1 = 23

Therefore, there are 23 integers between 200 and 500 (inclusive) that leave a remainder of 1 when divided by 7 and a remainder of 3 when divided by 4.

2) Finding the smallest positive integer that satisfies the given conditions:
To find the smallest positive integer that leaves a remainder of 5 when divided by 7, a remainder of 6 when divided by 11, and a remainder of 4 when divided by 13, we need to find the smallest number that satisfies each individual condition.

To satisfy the first condition (remainder of 5 when divided by 7), we can start from 5 and increment by 7 until it satisfies the condition.

5 mod 7 = 5 (satisfies!)

So, the smallest number that satisfies the first condition is 5.

To satisfy the second condition (remainder of 6 when divided by 11), we can start from where we left off (5) and increment by 77 (since 7 multiplied by 11 gives us 77) until it satisfies the condition.

5 + 77 = 82 mod 11 = 4 (does not satisfy)
5 + 2 * 77 = 159 mod 11 = 4 (does not satisfy)
5 + 3 * 77 = 236 mod 11 = 3 (does not satisfy)
5 + 4 * 77 = 313 mod 11 = 2 (does not satisfy)
5 + 5 * 77 = 390 mod 11 = 1 (does not satisfy)
5 + 6 * 77 = 467 mod 11 = 0 (does not satisfy)
5 + 7 * 77 = 544 mod 11 = 10 (does not satisfy)
5 + 8 * 77 = 621 mod 11 = 9 (does not satisfy)
5 + 9 * 77 = 698 mod 11 = 8 (does not satisfy)
5 + 10 * 77 = 775 mod 11 = 7 (satisfies!)

So, the smallest number that satisfies the second condition is 5 + 10 * 77 = 775.

To satisfy the third condition (remainder of 4 when divided by 13), we can start from where we left off (775) and increment by 77 * 11 (since 7 * 11 * 13 gives us 1001) until it satisfies the condition.

775 + 77 * 11 = 1542 mod 13 = 9 (does not satisfy)
775 + 77 * 2 * 11 = 3075 mod 13 = 4 (satisfies!)

So, the smallest number that satisfies all three conditions is 775 + 77 * 2 * 11 = 3075.

Therefore, the smallest positive integer that leaves a remainder of 5 when divided by 7, a remainder of 6 when divided by 11, and a remainder of 4 when divided by 13 is 3075.