Grades 11-12 Video Solutions 2024-2024_11-12

Video Transcription

A special 4-digit number, ABCD, satisfies the equation ABCD equals A to the A plus B to the B plus C to the C plus D to the D. What is the value of A?  The only digit for which N to the N will yield a 4-digit number is N equals 5. We have 5 to the 5 is 3125, 4 to the 4 is 256, and 6 to the 6 is greater than 10,000.  So exactly one of the digits must be 5 in order for our total to be a 4-digit number. We know that A equals 3 because the number must be at least 3,125, but it is not possible  for the number to be 4,000 or greater, since 5 to the 5 plus 4 to the 4 plus 4 to the 4 plus 4 to the 4 is less than 4,000, so A must be 3.  Then continuing in this way, we can find that A equals 3, B equals 4, C equals 3, and D equals 5, and our special number is 3435. Our answer is B.

Video Summary

A special 4-digit number, ABCD, satisfies the equation ABCD equals A to the A plus B to the B plus C to the C plus D to the D. What is the value of A? The only digit for which N to the N will yield a 4-digit number is N equals 5. We have 5 to the 5 is 3125, 4 to the 4 is 256, and 6 to the 6 is greater than 10,000. So exactly one of the digits must be 5 in order for our total to be a 4-digit number. We know that A equals 3 because the number must be at least 3,125, but it is not possible for the number to be 4,000 or greater, since 5 to the 5 plus 4 to the 4 plus 4 to the 4 plus 4 to the 4 is less than 4,000, so A must be 3. Then continuing in this way, we can find that A equals 3, B equals 4, C equals 3, and D equals 5, and our special number is 3435. Our answer is B.