PK 2019 Q15
15. The sum of the seven digits of the number ‘\(aaabbbb\)’ is equal to the two-digit number ‘\(ab\)’. What is the value of \(a + b\)?
- A 8
- B 9
- C 10
- D 11
- E 12
Pink Kangaroo 2019, Question 15. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: C 10
The value of ‘\(ab\)’ is \(10a + b\) so \(3a + 4b = 10a + b\) which gives \(3b = 7a\). Hence \(a\) is a multiple of 3 and \(b\) is a multiple of 7. Also \(a\) is non-zero (as it is the leading digit), and hence \(b\) also cannot be zero. The only single-digit solution of \(3b = 7a\) is \(a = 3\) and \(b = 7\), so \(a + b = 10\).
Solution © UK Mathematics Trust, from the PK 2019 solutions.