Mathematical Challenge for Filipino Kids Training Program

What they mean:
- Number: This is the idea of a quantity, like how many toys you have. It's an amount.
- Numeral: This is the symbol or picture we write for a number. Like the symbol "4" is a numeral that shows the number four.
Think about it like this: Your name (the numeral) is how people talk about you (the number). Different names can mean the same person, just like "four" and "4" mean the same amount.

Numbers and numerals are super important when we do math problems.
Things we look at: Whole numbers, numbers with decimal points, fractions, how to share things equally (divisibility), even and odd numbers, how decimal points move, and the very last digit of a number.
Ways to solve problems:
- Comparing things
- Choosing the right answers
- Finding examples that don't work
- Getting rid of wrong answers
- Mixing different ways to solve problems.
Why solving problems helps:
- Makes your brain better at thinking logically and doing calculations.
- Helps you remember and understand math rules, definitions, and properties you've learned before.

Problem: Imagine a secret number. If you move its decimal point one spot to the left, the new number is 40.68 smaller than the original secret number. What was the original secret number?
How to solve:
- Let's call the original secret number $x$ .
- When you move the decimal point one spot to the left, it's like dividing by 10. So the new number is $0.1x$ .
- The problem says the difference between them is 40.68.
- So, original number - new number = 40.68.
- $x - 0.1x = 40.68$
- $0.9x = 40.68$ (Because $1x - 0.1x = 0.9x$ )
- To find $x$ , we divide 40.68 by 0.9: $x = \frac{40.68}{0.9} = 45.2$ .
- So, the original number was 45.2.

Problem: You have a two-digit number (like 25). If you put a zero in the middle of its digits (e.g., 25 becomes 205), the new number is 8 times bigger than the original two-digit number. What was the two-digit number?
How to solve:
- Let the two-digit number be like $10 \times \text{first digit} + \text{second digit}$ . We'll use 'a' for the first digit and 'b' for the second.
- So the original number looks like $10a + b$ .
- When you put a zero in the middle, the new number looks like $100a + b$ .
- The problem says: New number = $8$ times original number.
- So, $100a + b = 8(10a + b)$
- Let's multiply out the right side: $100a + b = 80a + 8b$
- If we move the 'a's to one side and 'b's to the other, we get $10a=8b$ .
- From this, we look for digits 'a' and 'b'. The note states that the digits that work are $a = 4$ and $b = 5$ .
- So the two-digit number is 45.

Problem: You have a four-digit number. If you subtract the sum of its digits (add up all its digits) from the number, you get 658. Can you find a digit that is part of this number?
How to solve:
- Let the four-digit number be made of digits $a, b, c, d$ . So it's written as $1000a + 100b + 10c + d$ .
- The sum of its digits is $a + b + c + d$ .
- The problem says: Number - Sum of digits = 658.
- $(1000a + 100b + 10c + d) - (a + b + c + d) = 658$
- This simplifies to: $999a + 99b + 9c = 658$ .
- By using rules about what numbers can divide other numbers, we can figure out that the correct digit needed to solve this problem is 8.

What it is: The units digit is the very last digit on the right side of any number. For example, in the number 576, the units digit is 6.

What they mean: When you divide one number by another, sometimes there's a part left over that can't be divided perfectly. That leftover part is called the remainder.
Knowing about the units digit and remainders helps us understand how numbers work in math problems.

How to find it for big numbers: We can find the units digit of very large numbers (like when you multiply a number by itself many, many times) by looking for a pattern. The units digits often repeat in a cycle!

Problem: What is the units digit of $4^{993}$ ? (This means $4 \times 4 \times 4$ … 993 times!)
How to solve:
- Let's look at the units digits when we multiply 4 by itself:
- $4^1 = 4$ (The last digit is 4)
- $4^2 = 16$ (The last digit is 6)
- $4^3 = 64$ (The last digit is 4)
- $4^4 = 256$ (The last digit is 6)
- Do you see the pattern? It goes 4, 6, 4, 6…
- If the little number on top (the power) is odd, the units digit is 4.
- If the power is even, the units digit is 6.
- Our power is 993, which is an odd number.
- So, the units digit of $4^{993}$ is 4.

Problem: What is the units digit of the answer when you multiply $7295 \times 3158$ ?
How to solve:
- We only need to look at the last digit (units digit) of each number.
- The units digit of 7295 is 5.
- The units digit of 3158 is 8.
- Now, multiply just these units digits: $5 \times 8 = 40$ .
- The units digit of 40 is 0.
- So, the units digit of the big answer $7295 \times 3158$ is 0.

Numbers that divide 213 and leave 3 leftover: We learned about finding numbers that divide 213 but leave a remainder of 3. There are 7 such numbers!

Problems where you move decimal points around to find new numbers.
Figuring out differences and properties based on digits and numbers using rules from the examples.

Doing these kinds of problems helps you become a master of units digits, divisibility rules (which numbers can divide others perfectly), and thinking clearly to solve math puzzles!