Show that a positive integer is divisible by 11 if and only...

Name: Problem 32, Chapter 4: Discrete Mathematics and its Applications
Uploaded: 2020-06-22T20:42:09-08:00
Duration: 3 min 3 s
Channel: Ibrahima Barry
Description: Show that a positive integer is divisible by 11 if and only if the difference of the sum of its decimal digits in even-numbered positions and the sum of its decimal digits in odd-numbered positions is divisible by 11 .

Show that a positive integer is divisible by 11 if and only if the difference of the sum of its decimal digits in even-numbered positions and the sum of its decimal digits in odd-numbered positions is divisible by 11 .

Key Concepts

Powers of 10 Modulo 11

This concept highlights how the powers of 10 behave under modulo arithmetic, specifically modulo 11. The repeating pattern that emerges from powers of 10 modulo 11 underpins the divisibility rule for 11, as it explains why the difference between sums of alternating digit groups corresponds to the remainder when the number is divided by 11.

Positional Number System

This concept refers to the way numbers are expressed based on the position of each digit in a numeral, with each position representing a power of the base (in this case, base 10). It is fundamental in understanding how individual digits contribute to the overall value of a number, which is critical when analyzing digit positions for divisibility tests.

Modular Arithmetic

Modular arithmetic involves arithmetic with remainders, making it a central tool in number theory. In the context of divisibility, it helps simplify calculations by reducing numbers to their remainders when divided by a specific modulus. This is especially useful for understanding why certain patterns in digits, like the alternating sum in the divisibility rule for 11, determine divisibility.

Divisibility Rules

Divisibility rules provide quick methods to determine whether a number can be divided by another number without performing full division. These rules, which are often based on the digits of the number and their positions in a positional number system, simplify the process of identifying factors and understanding the underlying properties of numbers.

Transcript

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