A silly thing

Number Theory Level 2

Are there distinct positive integers m , n m,n such that:

  • m m and n n are 7-digit numbers.
  • m m and n n are formed by the digits 0 , 1 , 2 , 3 , 4 , 5 , 6 0,1,2,3,4,5,6 without repetition.
  • m m is a multiple of n n .
No Yes

Khang Nguyen Thanh by Khang Nguyen Thanh , 27, Vietnam

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1 solution

4102536 = 1025634 × 4 \Large {\color{#D61F06}4102536}={\color{#3D99F6}1025634}\times 4 The answer is YES .

Step-by-step approach:

  • 1 < m n ≤ 6 1<\dfrac{m}{n}\le 6 and m ≡ n ≡ 0 + 1 + 2 + 3 + 4 + 5 + 6 ≡ 3 ( m o d 9 ) m\equiv n\equiv 0+1+2+3+4+5+6\equiv 3\pmod{9} , so m = 4 n m=4n .
  • The first digit of n n must be 1.
1 Helpful 0 Interesting 0 Brilliant 0 Confused

How can we solve for this number, other than trial and error?

Calvin Lin Staff - 4 years ago

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Yeah, what if the number we take is 403561 or others. We need to find a general method which will always hold true.

Kaushik Chandra - 4 years ago

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