How do you solve these types of problems?
What is the remainder when 732^500 is divided by 10?
The remainder of a division problem is the same as its modulo.
732^500 % 10 = the remainder
what is modulo? and what do I need to do to solve this?
Modulo, denoted by the symbol "%", is a mathematical operation that returns the remainder when one number is divided by another. In this case, we can use modulo to find the remainder when 732^500 is divided by 10.
To solve this problem, we need to break it down into steps:
Step 1: Analyze the divisor (10) and find a pattern in its remainders.
- When any number is divided by 10, the remainder is simply the last digit of the number. For example:
- 27 % 10 = 7, because 27 has a remainder of 7 when divided by 10.
- 105 % 10 = 5, because 105 has a remainder of 5 when divided by 10.
Step 2: Determine the pattern of remainders for the given number (732) when divided by 10.
- Divide 732 by 10 and record the remainder: 732 % 10 = 2.
- Divide 73 (ignoring the last digit) by 10 and record the remainder: 73 % 10 = 3.
- Divide 7 (ignoring the last two digits) by 10 and record the remainder: 7 % 10 = 7.
Step 3: Observe the pattern that emerges from the remainders.
- As we saw in Step 2, the remainder pattern for 732 when divided by 10 is 7, 3, 2, 7, 3, 2, and so on.
Step 4: Find a pattern in the exponent (500).
- Notice that 500 is divisible by 3 since 500 ÷ 3 = 166 remainder 2.
Step 5: Apply the pattern to find the remainder.
- Since we observed the remainder pattern for 732 divided by 10 is 7, 3, 2, and it repeats, we can conclude that 732^500 has the same remainder as 732^2 when divided by 10.
- Calculate 732^2: 732^2 = 536,424.
- Find the remainder when 536,424 is divided by 10: 536,424 % 10 = 4.
Therefore, the remainder when 732^500 is divided by 10 is 4.