GK 2019 Q7
7. A four-digit integer is written on each of three pieces of paper and the pieces of paper are arranged so that three of the digits are covered, as shown. The sum of the three four-digit integers is 10 126. What are the covered digits?
- A 5, 6 and 7
- B 4, 5 and 7
- C 4, 6 and 7
- D 4, 5 and 6
- E 3, 5 and 6
Grey Kangaroo 2019, Question 7. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: A 5, 6 and 7
The required sum can be written as shown below, with \(a\), \(b\) and \(c\) as the missing digits:
The sum of the digits in the units column is 16 and hence there is a carry of 1 to the tens column. Therefore, when we consider the tens column, we have \(4 + a + 2 + 1 = 2\) or \(2 + 10\). Hence \(7 + a = 2\) or 12 and, since \(a\) is a positive single-digit integer, \(a = 5\) and there is a carry of 1 to the hundreds column. Similarly, when we consider the hundreds column, we have \(2 + 1 + c + 1 = 1\) or 11 and hence \(c = 7\) and there is a carry of 1 to the thousands column. Finally, when we consider the thousands and ten thousands columns, we have \(1 + 2 + b + 1 = 10\) and hence \(b = 6\). Therefore the missing digits are 5, 6 and 7.
Solution © UK Mathematics Trust, from the GK 2019 solutions.