A two-digit number is 7 times the sum of its two digits. The number that is formed by reversing its digits is 18 less than the original number. What is the number?

### Concept & Formula Algebraic relationships between a number, the sum of its digits, and its reversed form. When a number is reversed, the difference is always a multiple of 9: $9(x - y)$. ### Step-by-Step Solution Let the number be $10x + y$. Condition 2: The reversed number ($10y + x$) is 18 less than the original. $(10x + y) - (10y + x) = 18$ $9x - 9y = 18$ $9(x - y) = 18$ $x - y = 2$ Condition 1: The number is 7 times the sum of its digits. $10x + y = 7(x + y)$ $10x + y = 7x + 7y$ $3x = 6y$ $x = 2y$ Substitute $x = 2y$ into $x - y = 2$: $2y - y = 2$ $y = 2$ Now find $x$: $x = 2(2) = 4$ The number is $10(4) + 2 = 42$. ### Exam Strategy & Shortcut Option elimination is much faster here. Test condition 2 (reversing digits is 18 less) first: (a) $42 - 24 = 18$. (Matches! Check condition 1: $4+2=6$, and $7 \times 6 = 42$. Perfect.) (b) $52 - 25 = 27 \neq 18$. (c) $62 - 26 = 36 \neq 18$. (d) $72 - 27 = 45 \neq 18$. ### Common Pitfall Spending too much time setting up simultaneous equations when quickly checking the reversal difference for each option gives the answer in under 10 seconds. ### Final Answer Therefore, the correct answer is **42**.