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Find the sum of all the two digit numbers which can be expressed as an integral multiple of the product of their digits.

 Dec 8, 2019
 #1
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a=1;b=1;p=0; cycle:d=a*10+b;if(d%(a*b)==0, goto loop, goto next); loop:printd,", ",;p=p+n; next:b++;if(b<10, goto cycle, 0);b=1;a++;if(a<10, goto cycle, 0);print"Total = ",p

 

11 , 12 , 15 , 24 , 36 , Total = 98

 Dec 9, 2019
 #2
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Find the sum of all the two digit numbers which can be expressed as an integral multiple of the product of their digits.

 

I did this by looking at the numbers. It is not as arduous as it looks

 

10a+b=kab       where  1<= a <= 9       0<=b<=9     and k is a positive integer

 

If a=1 then

10+b=1b * k

b=0 no

b=1   k=11  yes   11

b=2    k=6  yes    12

b=3    no

b=4    14 is not a multiple of 4 so no

b=5     15 is a multple of 5  k= 3         15

b=6    16 is not a multiple of 6 so no

b=7     17 is not a multiple of 7 so no

b=8     18 is not a multiple of 8 so no

b=9     19 is not a multiple of 9 so no  

 

Ok now I can see a bit of a pattern emerging.  The original number must be a multiple of the last digit.

so when I look at a=2  I only need think about even 2 digit numbers

10a+b=kab 

If a=2 then

20+b= 2b * k

b=2  no

b=4    24 is divisable by 2*4=8  good    24

b=6  no, b=8 no

 

10a+b=kab    a=3

30+b=3b * k

If a=3 I only need consider numbers that are multiples of 3 which are  33,36,39

divide by 3  and I get   11,12,13

these must also be multiples of by b  

11/3  no,   12/6 yes  13/9 no

SO only   36 works here

 

10a+b=kab    a=4

40+b=4b * k

If a=4 I only need consider numbers that are multiples of 4 which are  44 and 48

divide by 4  and I get   11 and 12

these must also be multiples of by b  

11/4 no   12/8 no

SO none of these work

 

10a+b=kab    a=5

50+b=5b * k

only 55 is a multiple of 5

55/5=11   11 is not a multiple of 5 so no good

 

10a+b=kab    a=6

66 does not work

nor will 77 or 88 or 99

 

So the only ones that work are     11,  12,  15,  24  and 36.

 

Guest has already added them for you.

 Dec 9, 2019
edited by Melody  Dec 9, 2019

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