Criteria of divisibility by 9

Criteria of divisibility by 11

In what follows I shall be using representation of a number in various base systems. In the decimal (base 10) system a number a is written as

Generally speaking, for an integer base b the same number a will be written as

Notation (a)_b is meant to underscore the base in which a is expanded. The most customary is the decimal system. But, say, the French still use remnants of the system in base 20. With the advent of computers the binary (base 2) and hexadecimal (base 16) have been thrust into prominence. The octal system (base 8) is very useful in handling lengthy binary representations.

Theorem

Let b be an integer. Then an integer a

1. is divisible by (b-1) if and only if the sum of digits in its expansion (a)_b is divisible by (b-1)
2. is divisible by (b+1) if and only if the alternating sum of digits in its expansion (a)_b is divisible by (b+1)

Proof

For divisibility by 7 there are two criteria:

1. a = (a)₆ = c_k6^k+c_k-16^k-1+...+c₁6¹+c₀ is divisible by 7 iff the alternating sum of digits (-1)^kc_k+(-1)^k-1c_k-1+...+c₀ is divisible by 7.
2. a = (a)₈=c_k8^k+c_k-18^k-1+...+c₁8¹+c₀ is divisible by 7 iff the sum of digits c_k+c_k-1+...+c₀ is divisible by 7.

Example 2

512=8³. Therefore (512)₁₀=(1000)₈. This implies that divided by 7 the remainder will be 1. Then, for example, 511 is divisible by 7.

1296=6⁴. Therefore (1296)₁₀=(10000)₆. From here (1296+6)₁₀=(10010)₆. Its alternating sum of digits is 1-0+0-1+0=0. As a result, 1302 is divisible by 7.

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