Chung Kevin's favorite - (1)

Algebra Level 1

142857 × 2 = 285714 142857 × 3 = 428571 142857 × 4 = 571428 142857 × 5 = 714285 142857 × 6 = 857142 142857 \times 2 = 285714 \\ 142857 \times 3 = 428571 \\ 142857 \times 4 = 571428 \\ 142857 \times 5 = 714285 \\ 142857 \times 6 = 857142 \\

You can see that the multiples of 142857 you would notice that they are just the permutation of digits 1,4,2,8,7,5.

Can you compute 142857 × 7 142857 \times 7 ?


The answer is 999999.

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Nihar Mahajan by Nihar Mahajan , 20, India

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4 solutions

Chew-Seong Cheong
May 30, 2015

Why use 2 + 5 2+5 when we can use 6 + 1 6+1 .

142857 × 7 = 142857 × ( 6 + 1 ) = 857142 + 142857 = 999999 142857\times 7 = 142857\times (6+1) = 857142+142857 = \boxed{999999} .

18 Helpful 1 Interesting 0 Brilliant 0 Confused
Siddharth Singh
May 30, 2015

The expression can be written as \text{The expression can be written as}

142857*(2+5)=285714+714285=999999 \text{142857*(2+5)=285714+714285=999999}

11 Helpful 0 Interesting 0 Brilliant 0 Confused

Moderator note:

Yes, a simple 7 = 1 + 6 7 = 1 + 6 or 7 = 2 + 5 7 = 2 + 5 or even 7 = 3 + 4 7 = 3 + 4 solves this problem.

Bonus question: What is so special about the number 142857 that every time you multiple it by an integer between 1 to 6, the resultant number is the rearrangement of digits of the original number? Can you find other numbers that share the same property?

In response to Challenge Master :Such numbers are known as Cyclic Numbers .

Rohit Ner - 6 years ago

Another Interesting thing to note— 1/7 = 0.142857142857142857... (repeating set of 142857)

Yogesh Verma - 6 years ago

@Chung Kevin did you see this?

Nihar Mahajan - 6 years ago

Log in to reply

Yes I did! I'm wondering how many people got it wrong, and why.

Chung Kevin - 6 years ago

Note that:

142+857=999

285+714=999

428+571=999 ...

and that:

14+28+57=99

42+85+71=2(99)

You can rearrange the numbers to get the others.

Also, (14)(28)(57)+(42)(85)(71)=275814, rearranged yet another way!

Anthony Pham - 5 years, 11 months ago

52631578947368421 works up to 19. If you're curious enough, you'll find that 52631578947368421 x 19 = 999999999999999999.

Anthony Pham - 5 years, 11 months ago
Tj Dychitan
Dec 31, 2015

Since,

142857 x 7 = 142857 x 6 + 142857

Substituting the expression, we'll have,

142857 x 7 = 857142 +142857

142857 x 7 = 999999 \boxed{999999}

1 Helpful 1 Interesting 1 Brilliant 0 Confused

142857×7 = 142857× (10-3)= = (142857×10) - (142857×3)= = 1428570 - 428571= 999999! It was quite easy!

0 Helpful 0 Interesting 0 Brilliant 0 Confused

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