Find the least multiple of 23 which when divided by 24, 21, and 18 leaves the remainders 13, 10, and 7 respectivelyOptions :A] 3004B] 3024C] 3013D] 3026

To solve this problem, we need to find a number that fits all the given conditions.

Step 1: Find the least common multiple (LCM) of 24, 21, and 18. The LCM of these numbers is 504.

Step 2: The number we are looking for is a multiple of 23 and leaves a remainder of 13 when divided by 24, 10 when divided by 21, and 7 when divided by 18. This means the number is of the form 504k + 13, 504k + 10, and 504k + 7.

Step 3: To find the least such number, we can start checking from k = 1. We find that 504*1 + 13 = 517 is not divisible by 23.

Step 4: We continue this process until we find a number that is divisible by 23. We find that 5046 + 13 = 3029 is not divisible by 23, but 5046 + 10 = 3026 is divisible by 23.

So, the least multiple of 23 which when divided by 24, 21, and 18 leaves the remainders 13, 10, and 7 respectively is 3026.

Therefore, the answer is D] 3026.

To solve this problem, we need to find a number that fits all the given conditions.

Step 1: Since the remainders when the number is divided by 24, 21, and 18 are 13, 10, and 7 respectively, we can say that the number is of the form 24x+13, 21y+10, and 18z+7, where x, y, and z are integers.

Step 2: We notice that the differences between the divisors and the remainders are the same (24-13=21-10=18-7=11). So, the number is also of the form 11n+10, where n is an integer.

Step 3: The least multiple of 23 which is of the form 11n+10 can be found by checking multiples of 23 one by one. We find that 23*131=3003 is of the form 11n+10.

Step 4: However, we need a number that is 1 more than this (because the remainder when divided by 24, 21, and 18 should be 13, 10, and 7, not 12, 9, and 6). So, the least such multiple of 23 is 3003+1=3004.

So, the answer is A] 3004.

The problem is asking for the least common multiple (LCM) of 23 that leaves a remainder of 13 when divided by 24, a remainder of 10 when divided by 21, and a remainder of 7 when divided by 18.

Step 1: Find the LCM of 24, 21, and 18. The LCM of 24, 21, and 18 is 504.

Step 2: The numbers 504k+13, 504k+10, and 504k+7 should be the same for some integer k, because they are all multiples of 23.

Step 3: We can see that the difference between the remainders is the same (3), so we can write the number we are looking for as 504k+13.

Step 4: Now we need to find the smallest k such that 504k+13 is a multiple of 23. We can do this by trying out different values of k.

Step 5: If we try k=1, we get 517 which is not a multiple of 23. If we try k=2, we get 1021 which is not a multiple of 23. If we continue this process, we find that k=6 gives 3025 which is not a multiple of 23, but k=7 gives 3533 which is also not a multiple of 23.

Step 6: If we continue this process, we find that k=12 gives 6053 which is not a multiple of 23, but k=13 gives 6557 which is also not a multiple of 23.

Step 7: If we continue this process, we find that k=59 gives 29737 which is a multiple of 23.

Step 8: So, the least multiple of 23 which when divided by 24, 21, and 18 leaves the remainders 13, 10, and 7 respectively is 29737.

However, none of the options (A: 3004, B: 3024, C: 3013, D: 3026) match this result. There seems to be a mistake in the problem or the options provided.

To solve this problem, we need to find a number that when divided by 24, 21, and 18 leaves the remainders 13, 10, and 7 respectively.

First, we need to find a number that when divided by 24, 21, and 18 leaves the same remainder. Since the differences between the divisors and the remainders are the same (24-13=11, 21-10=11, 18-7=11), we can say that the number we are looking for is a multiple of the least common multiple (LCM) of 24, 21, and 18, plus the common difference 11.

The LCM of 24, 21, and 18 is 504. So, the number we are looking for is of the form 504k + 11.

Now, we need to find the least multiple of 23 that is of the form 504k + 11. We can do this by checking the multiples of 23 one by one until we find a number that is of the form 504k + 11.

After checking the multiples of 23, we find that the least multiple of 23 that is of the form 504k + 11 is 3013.

So, the answer is C] 3013.