Determining Divisibility by Digits

Sometimes, by looking at the digits of numbers that have at least 4 digits, you can tell that it's divisible by other specific numbers! (Decimal points unconsidered, meaning that we're only going to consider integers!)

Examples:

3417 is divisible by 17 because:

3417 ÷ 17 = 201

Remember this fact about division: The dividend is also divisible by the quotient, not just the divisor! In fact, the divisor & the quotient can switch numbers without falsifying the math statement!

1414 is divisible by 14 because:

1414 ÷ 14 = 101

Any 4-digit number that reads: "ABAB" is divisible by 101. A is the 1st digit & B is the 2nd digit. (And of course A & B can be the same digit!)

I just remembered that when I type integers with more than 3 digits, I usually use commas for digit counting. However, I made an exception with the 1st 2 examples. The following examples will include integers with more than 4 digits! (A comma is placed after every 3 digits. They're called thousands seperators.)

369,123 is divisible by 3,001 because:

369,123 ÷ 3,001 = 123

145,145 is divisible by 1,001 because:

145,145 ÷ 1,001 = 145

24,682,468 is divisible by 2,468 because:

24,682,468 ÷ 2,468 = 10,001

149,298 is divisible by 1,002 because:

149,298 ÷ 1,002 = 149

333,222,111 is divisible by 111 because:

333,222,111 ÷ 111 = 3,002,001

999,999,999 is divisible by 999 because:

999,999,999 ÷ 999 = 1,001,001

1,428,570,142,857 is divisible by 142,857 because:

1,428,570,142,857 ÷ 142,857 = 10,000,001

Wait a minute! Have you noticed something about most of these examples? Most of the divisors or quotients have at least 1 zero or one as a digit! The dividend in the final example has a zero for a very special reason! Most numbers that have digit patterns like these are likely to be divisible by numbers such as 11, 101, 111, 1,001, etc. It also helps to know the multiples of those specific numbers you see in the patterns!

This math trick kind of reminds me of the number sandwiches from 2 older math tricks of mine! They're #53 & #54, entitled: "Digit Cloning With Multiplication" & "Number Sandwiches" respectively!

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