When a number is divided by 280, the remainder is 73. What will be the remainder when the same number is divided by 35?

Steps to Solve

1. Determine the Original Number

Let the original number be $x$.

According to the problem, when $x$ is divided by 280, the remainder is 73. This can be expressed mathematically as: $$ x = 280k + 73 $$ where $k$ is the quotient (an integer).

2. Find the Value of k within a Range

Next, we need possible values of $x$ that are less than 280 (since $k$ will vary). The equation simplifies to: $$ x = 280k + 73 $$ For our purposes, we will consider $k = 0$. Thus, $$ x = 0 \cdot 280 + 73 = 73 $$

3. Check if k can be greater

We can also consider $k = 1$: $$ x = 1 \cdot 280 + 73 = 353 $$

Since for $k = 1$, $x$ exceeds 280, we will use $x = 73$ to find the remainder when divided by 35.

4. Calculate Remainder when Divided by 35

Now we calculate the remainder when $x = 73$ is divided by 35: $$ 73 \div 35 = 2 \text{ (quotient)} $$ $$ \text{Remainder} = 73 - (2 \cdot 35) = 73 - 70 = 3 $$

Thus, for completeness, let's verify with $x = 353$: $$ 353 \div 35 = 10 \text{ (quotient)} $$ $$ \text{Remainder} = 353 - (10 \cdot 35) = 353 - 350 = 3 $$

So, both cases yield the same remainder.