Consecutive numbers:
Consecutive numbers are numbers that follow each other in order without any gaps, where each number is exactly \(1\) more than the previous number. For example: \(5, 6, 7, 8\) are consecutive numbers.
General form of consecutive numbers:
Any consecutive number can be written as \(x\), (\(x + 1\)), (\(x + 2\)), (\(x + 3\)), ....
Any even consecutive number can be written as \(2x\), \(2x + 2\), \(2x + 4\), ....
Any odd consecutive number can be written as \(2x+1\), \(2x + 3\), \(2x + 5\), ....
Magic of \(+\) and \(-\) with \(4\) consecutive numbers:
Let's take four consecutive numbers as \(3\), \(4\), \(5\) and \(6\).
Now, we place \(+\) or \(-\) signs between them.
- \(3 + 4 + 5 + 6 = 18\)
- \(3 + 4 + 5 - 6 = 6\)
- \(3 + 4 - 5 + 6 = 8\)
- \(3 - 4 + 5 + 6 = 10\)
- \(3 - 4 - 5 + 6 = 0\)
- \(3 + 4 - 5 - 6 = -4\)
- \(3 - 4 + 5 - 6 = -2\)
- \(3 - 4 - 5 - 6 = -12\)
Here, note that sum of all the possibilities of consecutive numbers are even number.
Thus, sum of any four consecutive numbers is even.
What is Parity?
Parity tells us if a number is even or odd:
Parity tells us if a number is even or odd:
- An even number leaves no remainder when divided by \(2\). It is called as even parity (eg: \(6 \div 2 = 3\) with remainder \(0\)).
- An odd number leaves a remainder of \(1\) when divided by \(2\). It is called as odd parity (eg: \(7 \div 2 = 3\) with remainder \(1\)).
Addition and Subtraction Parity:
odd \(\pm\) odd \(=\) even
even \(\pm\) even \(=\) even
odd \(\pm\) even \(=\) odd
Product of Parity:
odd \(\times\) even \(=\) even
odd \(\times\) odd \(=\) odd
Exponent Parity:
odd \(^3 =\) odd
odd \(\pm\) odd \(=\) even
even \(\pm\) even \(=\) even
odd \(\pm\) even \(=\) odd
Product of Parity:
odd \(\times\) even \(=\) even
odd \(\times\) odd \(=\) odd
Exponent Parity:
odd \(^3 =\) odd
Digits in Disguise
Digits in disguise means letters or symbols are used instead of numbers (digits) in a math problem or puzzle. Each letter stands for one number from \(0\) to \(9\). The goal is to find which digit each letter represents.
It’s like a secret code where every letter hides a number, and you need to solve the puzzle by using clues and logic.
How to Solve Digits in Disguise?
- Remember each letter is one number, always the same number wherever it appears.
- Different letters stand for different numbers.
- Use simple math rules to guess which numbers fit.
- Check if the sum or product is right after you guess.
Example:
Find the value of \(A\) and \(B\).
Solution:
It is given that \(B \times B = 6\).
Thus, \(B\) takes the value either \(4\) or \(6\).
When \(B = 4\), we have:
\(\Rightarrow (A \times 4) + 1 = 9\) [Where \(1\) is the carry over of \(16\)]
\(\Rightarrow 4A + 1 = 9\)
\(\Rightarrow 4A = 8\)
\(\Rightarrow A = 2\)
Thus, \(A = 2\) and \(B = 4\).