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Number Play - Pretty Palindromic Patterns

Grade 6CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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A palindromic number is a number that reads the same forwards and backwards. Examples include 121121, 13311331, and 45544554.

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Any number can potentially be turned into a palindrome using the 'Reverse and Add' method. This involves taking a number, reversing its digits, and adding the result to the original number. This process is repeated until a palindrome is reached.

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Special patterns exist when multiplying numbers made of only ones. For example, squaring 1111, 111111, or 11111111 always results in a palindromic number.

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The middle digit of a square of nn ones (where n≤9n \le 9) is equal to nn, and the digits decrease on either side back to 11.

📐Formulae

Number+Reverse of Number=Palindrome (eventually)\text{Number} + \text{Reverse of Number} = \text{Palindrome (eventually)}

11×11=12111 \times 11 = 121

111×111=12321111 \times 111 = 12321

1111×1111=12343211111 \times 1111 = 1234321

💡Examples

Problem 1:

Use the 'Reverse and Add' method to turn the number 4848 into a palindrome.

Solution:

Step 1: Take the number 4848. Reverse it to get 8484. Step 2: Add them: 48+84132\begin{array}{r} 48 \\ + 84 \\ \hline 132 \end{array} Step 3: 132132 is not a palindrome. Reverse 132132 to get 231231. Step 4: Add them: 132+231363\begin{array}{r} 132 \\ + 231 \\ \hline 363 \end{array} Result: 363363 is a palindromic number.

Explanation:

We start with 4848. Since the first addition resulted in 132132 (not a palindrome), we repeated the process with 132132 to get 363363, which is a palindrome.

Problem 2:

Predict the product of 11111×1111111111 \times 11111 using the palindromic pattern.

Solution:

The number 1111111111 has 55 ones. Following the pattern:

  • 11 one: 1×1=11 \times 1 = 1
  • 22 ones: 11×11=12111 \times 11 = 121
  • 33 ones: 111×111=12321111 \times 111 = 12321
  • 44 ones: 1111×1111=12343211111 \times 1111 = 1234321 Therefore, for 55 ones: 11111×11111=12345432111111 \times 11111 = 123454321

Explanation:

When squaring a number consisting of nn ones, the digits increase from 11 up to nn and then decrease back to 11.

Problem 3:

Check if the sum of 273273 and its reverse is a palindrome.

Solution:

Original number: 273273 Reverse of number: 372372 Addition: 273+372645\begin{array}{r} 273 \\ + 372 \\ \hline 645 \end{array} Is 645645 a palindrome? No, because 645≠546645 \neq 546.

Explanation:

The sum of a number and its reverse does not always result in a palindrome in just one step. In this case, 645645 would need another 'Reverse and Add' step.