Strange four-digit number

Level pending

A four-digit number of the form a b c d abcd has a special property such that a + c = b + d a+c=b+d .The number n n is a divisor of all these numbers of this form.Also, n n is an odd prime positive integer.Find n n .


The answer is 11.

Lorenc Bushi by Lorenc Bushi , 21, Albania

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2 solutions

Jubayer Nirjhor
Dec 21, 2013

The question states that...

a + c = b + d ⟹ b + d − a − c = 0 a+c=b+d ~~~ \Longrightarrow ~ b+d-a-c=0

Now notice that...

a b c d ‾ = 1000 a + 100 b + 10 c + d \overline{abcd}=1000a+100b+10c+d

= ( 1001 a + 99 b + 11 c ) + ( b + d − a − c ) =(1001a+99b+11c)+(b+d-a-c)

= 1001 a + 99 b + 11 c + 0 =1001a+99b+11c+0

= 11 ( 91 a + 9 b + c ) =11(91a+9b+c)

Since 11 11 is an odd prime positive integer, we are done. Therefore...

n = 11 n=\fbox{11}

3 Helpful 0 Interesting 0 Brilliant 0 Confused
Thomas Crow
Jun 3, 2014

I decided to try it with the easiest two examples of these I could think of. These were 1342 and 4312. I then found the prime factors of both of these in turn.

For 1342: (/2) 671 (/11) 61. 61 is a prime. Prime factors of 1342 are: 2x11x61

For 4312: (/2) 2156 (/2) 1078 (/2) 539 (/7) 77 (/11) 7. 7 is a prime. Prime factors of 4312 are: 2^3x7x11

The two numbers share prime factors of 2 and 11. The question specifically refers to ODD prime factors, and as such...

THE ANSWER IS 11.

0 Helpful 0 Interesting 0 Brilliant 0 Confused

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