## Num3ers such xyz + x(10y + z) + (10x + y)z = 100x + 10y + z

$x y z \; + \; x (10 y + z) \; + \; (10 x + y) z \; = \; 100 x \; + \; 10 y \; + \; z$

for example,   155

$1 \times \: 5 \times \: 5 \; + \; 1 \times \: (10 \times \: 5 + 5) \; + \; (10 \times \: 1 + 5) \times \: 5 \; = \; 155$

Find a 4-digit number abcd such that

a*b*c*d   +   abc*d +   ab*cd +     a*bcd   =   abcd

Pipo found :

6550 = 6*5*5*0 + 6*550 + 65*50 + 655*0

13142 = 1*3*1*4*2 + 1* 3142 + 13 *142 + 131* 42 + 1314*2
14610 = 1*4*6*1*0 + 1* 4610 + 14 *610 + 146* 10 + 1461*0
16420 = 1*6*4*2*0 + 1* 6420 + 16 *420 + 164* 20 + 1642*0
23610 = 2*3*6*1*0 + 2* 3610 + 23* 610 + 236* 10 + 2361*0
34420 = 3*4*4*2*0 + 3* 4420 + 34* 420 + 344* 20 + 3442*0
65500 = 6*5*5*0*0 + 6* 5500 + 65* 550 + 655* 50 + 6550*0

146100 = 1*4*6*1*0*0 + 1* 46100 + 14 *6100 + 146* 100 + 1461*0+ 14610*0
164200 = 1*6*4*2*0*0 + 1* 64200 + 16 *4200 + 164* 200 + 1642*0+ 16420*0
236100 = 2*3*6*1*0*0 + 2* 36100 + 23* 6100 + 236* 100 + 2361*0+ 23610*0
344200 = 3*4*4*2*0*0 + 3* 44200 + 34* 4200 + 344* 200 + 3442*0+ 34420*0
655000 = 6*5*5*0*0*0 + 6* 55000 + 65* 5500 + 655* 500 + 6550*0+ 65500*0

A pattern seems to emerge

155    –>    6550    –>    65500    –>    655000

13142
16420
34420
164200
344200

14610
23610
146100
236100

math grad - Interest: Number theory
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### 2 Responses to Num3ers such xyz + x(10y + z) + (10x + y)z = 100x + 10y + z

1. pipo says:

Actually in the ‘1545’-example, one item is missing. In the example it says abcd + abc*d+ ab*cd or better a*b*c*d + (100*a+10*b+c)*d + (10*a+b)*(10*c+d) = 1000*a+ 100*b + 10*c +d
There are six numbers between 1000 and 10000 that satisfy:
4563 = 4*5*6*3 + 456*3 + 45*63
4345 = 4*3*4*5 + 434*5 + 43*45
4318 = 4*3*1*8 + 431*8 + 43*18
1545 as above
1518 = 1*5*1*8 + 151*8 +15*18
1064 = 1*0*6*4 + 106*4 +10*64

Better would be: a*b*c*d + (100*a+10*b+c)*d + (10*a+b)*(10*c+d) +a*(100*b +10*c+ d)= 1000*a+ 100*b + 10*c +d.

Only one number between 1000 and 10000 satisfies:

6550 = 6*5*5*0 + 6*550 + 65*50 + 655*0

Between 10000 and 100000 we have:

13142 = 1*3*1*4*2 + 1* 3142 + 13 *142 + 131* 42 + 1314*2
14610 = 1*4*6*1*0 + 1* 4610 + 14 *610 + 146* 10 + 1461*0
16420 = 1*6*4*2*0 + 1* 6420 + 16 *420 + 164* 20 + 1642*0
23610 = 2*3*6*1*0 + 2* 3610 + 23* 610 + 236* 10 + 2361*0
34420 = 3*4*4*2*0 + 3* 4420 + 34* 420 + 344* 20 + 3442*0
65500 = 6*5*5*0*0 + 6* 5500 + 65* 550 + 655* 50 + 6550*0
The last one is pretty obvious.

Between 100000 and 1000000 we have:

146100 = 1*4*6*1*0*0 + 1* 46100 + 14 *6100 + 146* 100 + 1461*0+ 14610*0
164200 = 1*6*4*2*0*0 + 1* 64200 + 16 *4200 + 164* 200 + 1642*0+ 16420*0
236100 = 2*3*6*1*0*0 + 2* 36100 + 23* 6100 + 236* 100 + 2361*0+ 23610*0
344200 = 3*4*4*2*0*0 + 3* 44200 + 34* 4200 + 344* 200 + 3442*0+ 34420*0
655000 = 6*5*5*0*0*0 + 6* 55000 + 65* 5500 + 655* 500 + 6550*0+ 65500*0
But they are all obvious.

Pipo

• benvitalis says:

Right. I didn’t specify. I should have asked to break up an integer into all possible groups (and NOT any possible, like I did) without changing the order of the digits of the integer. In that case, my second example, 1545, in not valid. Thanks.