Problem 24. ABCDEF is a 6-digit number made up of the digits 1, 2, 3, 4, 5, and 6, with no repeated digits. Its first two digits, AB, is a multiple of 2. Its first three digits, ABC, is a multiple of 3. Its first four digits, ABCD, is a multiple of 4. Its first five digits, ABCDE, is a multiple of 5. And the full number is a multiple of 6. What is the 6th digit, So, we'll look at each condition one by one. So, first, we know that AB is divisible by 2, ABCD is divisible by 4, and ABCDEF is divisible by 6. So since 4 and 6 are also divisible by 2, this means that AB, ABCD, and ABCDEF are all divisible by 2, so they're all even. And then we can tell a number is even based on its last digit. So that means that the last digits, B, D, and F, have to be even. And since we have three even digits, 2, 4, and 6, we know that B, D, and F have to be 2, 4, and 6 in some order. And this means that A, C, and E have to be 1, 3, and 5 in some order. Now, we can also figure out that E has to be 5, because we know that a number that's divisible by 5 ends in either 5 or 0. 0 is not an option here, so that means that the number has to end in 5, and so E is 5. Now, since A, C, and E were 1, 3, 5 in some order, and E is 5, this means that A and C are 1 and 3 in some order. Now let's look at the condition that ABC is a multiple of 3. So we know that a number is a multiple of 3 when the sum of its digits is a multiple of 3. So since ABC is a multiple of 3, A plus B plus C is divisible by 3. Now we know that A and C are 1 and 3 in some order, but no matter what the order is, A plus C is 4. And so we need to find a B such that B plus 4 is divisible by 3. And out of 2, 4, and 6, the only number that works here is 2. So now we know that B is 2. Now let's look at ABCD. It needs to be divisible by 4. The condition for divisible by 4 is that the last two digits are divisible by 4. So CD needs to be divisible by 4. Now C is either 1 or 3, and D is either 4 or 6. So 14 and 34 are both not divisible by 4, but 16 and 36 are both divisible by 4. So since C is odd, D has to be 6, D can't be 4. And so now we know that B is 2 and D is 6, which means that F, which is the last even number, has to be 4. And so the answer is B.

puzzle

divisibility

6-digit number

permutation

logic