<p>What is the remainder when \(5\times 55\times 555 \times \ldots \times \underbrace{555\ldots 55}_{555 5's}\) is divided by \(100\)? </p>
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i need answer on mth 121
To find the remainder when a large number is divided by another number, we can apply the concept of modular arithmetic.
First, let's consider the pattern of the number being multiplied:
- The first number is 5.
- The second number is obtained by appending another 5 to the right of the previous number: 55.
- The third number is obtained by appending another 5 to the right of the previous number: 555.
- This pattern continues with each subsequent number.
We can define the pattern recursively:
- Let's denote the first number as \(a_1 = 5\).
- The subsequent numbers can be defined as \(a_{n+1} = 10a_n + 5\).
Now, we need to find the remainder when the product of these numbers is divided by 100.
To simplify the calculation, we can take the remainder modulo 100 after each multiplication.
Let's calculate the remainders step by step:
- \(a_1 = 5\) (The remainder here is 5.)
- \(a_2 = 10a_1 + 5 = 10\times 5 + 5 = 55\) (The remainder here is 55.)
- \(a_3 = 10a_2 + 5 = 10\times 55 + 5 = 555\) (The remainder here is 55.)
- \(a_4 = 10a_3 + 5 = 10\times 555 + 5 = 5555\) (The remainder here is 55.)
- \(a_5 = 10a_4 + 5 = 10\times 5555 + 5 = 55555\) (The remainder here is 55.)
- And so on...
We notice that the remainder remains the same (55) as we continue the pattern.
So, the remainder when \(5\times 55\times 555 \times \ldots \times \underbrace{555\ldots 55}_{555 \, 5's}\) is divided by 100 is 55.