Question

What will be the highest three-digit number that, when divided by 3, 7, and 21, leaves a remainder of 2?

a. 996
b. 997
c. 998
d. 999

Answer 1

The largest three-digit number divisible by 3, 7, and 21 is one less than a multiple of their LCM, which is 21. The largest three-digit multiple of 21 is 993, and adding 2 gives us 995, which is not in the options. The correct number below 995 that fits the criteria is 974, which is not in the provided options.

Step-by-step explanation:

The question asks us to find the highest three-digit number that leaves a remainder of 2 when divided by 3, 7, and 21. The number that is divisible by these three numbers is a multiple of their least common multiple (LCM). Since 7 is a factor of 21, the LCM of 3, 7, and 21 is simply 21. Therefore, we are looking for the largest three-digit multiple of 21, and then we will add 2 to this number, to find the number that leaves a remainder of 2 when divided by these numbers.

Let's first find the largest multiple of 21 within the three-digit range:

1. Divide 999 (the largest three-digit number) by 21.
2. We get 47 as the quotient and 6 as the remainder.
3. Subtract the remainder from 999 to get the highest multiple of 21, which is 999 - 6 = 993.

Now, we add 2 to 993 to find the required number:

993 + 2 = 995.

However, 995 is not in the given options. The correct answer must be slightly less than 995 and divisible by 21, so we check the next lower multiple of 21:

1. Subtract 21 from 993 to get 972.
2. Then, add 2 to 972 to get 974, which is the highest number that leaves a remainder of 2 when divided by 3, 7, and 21.

None of the options (996, 997, 998, 999) given are correct.