If $x$ and $y$ are two digits of the number $653xy$ such that the number is divisible by 80, then $x + y$ is equal to

## Concept & Logic To solve for divisibility by a large composite number like 80, we break it down into its **co-prime factors**. The most effective co-prime factors of $80$ are $10$ and $8$. Therefore, for a number to be divisible by 80, it must satisfy both: 1. **Divisibility by 10:** The unit digit must strictly be `0`. 2. **Divisibility by 8:** The number formed by the last three digits must be divisible by `8`. ## Step-by-Step Solution **Given:** The five-digit number is $653xy$. It is completely divisible by 80. **Calculation / Deduction:** * **Step 1: Apply the rule for 10.** For $653xy$ to be divisible by 10, the unit digit must be zero. Therefore, $y = 0$. The number is now $653x0$. * **Step 2: Apply the rule for 8.** For $653x0$ to be divisible by 8, its last three digits ($3x0$) must be a multiple of 8. Let's test possible values for $x$ (from 0 to 9) to see which makes $3x0$ divisible by 8: * If $x = 2$, the number is $320$. $320 \div 8 = 40$ (Perfectly divisible). * If $x = 6$, the number is $360$. $360 \div 8 = 45$ (Perfectly divisible). * No other single digit $x$ yields a multiple of 8 (e.g., 300, 340, 380 are not divisible by 8). * **Step 3: Evaluate the required sum ($x + y$).** We have two possible scenarios for $(x, y)$: $(2, 0)$ and $(6, 0)$. * Scenario 1: $x + y = 2 + 0 = 2$. * Scenario 2: $x + y = 6 + 0 = 6$. Looking at the given options (3, 4, 5, 6), only $6$ is available. ## Exam Strategy & Shortcut Always address the rule that restricts the unit digit first. Seeing `80` instantly tells you $y = 0$. This reduces the problem to finding $x$ such that $3x0$ is divisible by 8. You can rapidly mentally estimate multiples of 80 near 300: $8 \times 40 = 320$ and $8 \times 45 = 360$. This instantly gives you $x = 2$ or $x = 6$. Add $0$ to either, and match with the options. ## Common Pitfall A common trap is breaking $80$ into factors that are not co-prime, such as $40$ and $2$, or $16$ and $5$. While 16 and 5 are co-prime, the divisibility rule for 16 requires checking the last four digits, which is unnecessarily complex. Using 10 and 8 is mathematically identical and much faster to execute. ## Final Answer **Therefore, the correct answer is 6.**