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When I an initial saw this puzzle, I immediately thought that a brute pressure approach; a couple of lines that code and you deserve to permute with every possible mix of the digits in a few seconds, yet by applying a pair of the basic divisibility rules we"re taught in schools, you deserve to solve this v pen, paper, and also a half a cup of coffee.

You are watching: Arrange the digits 1 through 9

If you need a refresher on an easy divisibility tests, girlfriend can uncover one here.

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Answers

Here is the Answer:

381654729

Derivation

ABCDEFGHI

First that all, there is a "gimmie". Together we"re utilizing the number 1-9, then whatever arrangement we select it will be divisible by nine. 1+2+3+4+5+6+7+8+9=45, which is divisible by nine. (A divisibility ascendancy for nine is that digit root is divisible by nine). So, we don"t treatment what digit goes at the end!

ABCDEFGHI

Next, we can apply the divisibility by five rule. Every number that is divisible by 5 has to end in a zero or five. As we"re not utilizing zero, the fifth digit has to be five.

ABCD5FGHIUsed: 1 2 3 4 5 6 7 8 9

Next, we recognize that, at minimum, digits: B,D,F,H have to be also 2,4,6,8 together these have to be divisible by also numbers; this reduces down the solution set but us need much more help. Native the divisibility of 3 rule, A+B+C must be divisible through 3, as have to D+5+F (all numbers divisible by 6 are likewise divisible by three).

Combing these, as we recognize D,F have to be from the set 2,4,6,8, and also that D+5+F requirements to be divisible by three, out of every the combinations, only four are possible: 254 256 258 452 456 458 652 654 658 852 854 856.

Out of the four solutions: 258 456 654 852 us can eliminate two the these v application the the divisibility by four test (To it is in divisible by four, the last 2 digits must likewise be divisible by four). So, for ABCD to be divisible through four, climate CD likewise needs to it is in divisible by four. Together we need four also digits in positions B,D,F,H this means that A,C,E,G,I should be odd. Because that CD to it is in divisible by four, and also with C being odd, then D cannot be 8 or 4. There room now just two possibilities: 258 456 654 852.

Let"s inspection both that these:

ABC258GHI

Inserting among the 2 remaining also digits in B,H 4,6, then making sure A+B+C add up to multiple of 3 leads to simply eight possible choices:

147258369147258963714258369714258963369258147369258741963258147963258741

The divisibility test because that eight is that the last three digits must be divisible through eight also. This rules out the arrangements finishing -836, -814, and -874.

147258369147258963714258369714258963369258147369258741963258147963258741

The continuing to be two fail the department test for seven digits. This route is a dead end with no solution.

147258369147258963714258369714258963369258147369258741963258147963258741
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ABC654GHI

The divisibility by eight check we used on the left (last 3 digits likewise divisible through eight), method than 4GH needs to be divisible by eight. H must be from the collection 8,2, however it can"t it is in 8 together G is odd; in fact G needs to be 3,7.

With these constraints, there room eight potential solutions:

987654321789654321381654729183654729981654327189654327981654723189654723

Manually applying the divisibility by 7 test, just one systems survives.

987654321789654321381654729183654729981654327189654327981654723189654723

We have a winner!

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This trouble has a distinctive solution: 381654729

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Cryptogram Puzzle

If you favored this puzzle, examine out a comparable one ns posted critical year whereby you have to arrange the exact same digits to complete an equation.


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