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# Number Theory Help.

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Of all 5-digit integers, how many of them are multiples of 2, 3, 4, 5, 6, 7, 8, 9 for each 5-digit integer? Thank you.

Dec 15, 2019

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The easiest and simplest way of  finding out the integers in question is simply to multiply the small primes by each other to see if you can get 5-digit numbers. That is : 2, 3, 5, 7. The problem is that it is time consuming, since you have to try many different exponents such as: 2^3 x 3^2 x 5 x 7^2 =17640, which is divisible by 2, 3, 4, 5, 6, 7, 8, 9.......etc.

Since, it would take some time to find them all, I wrote a small computer program to accomplish the same task, execept it takes the computer a few milliseconds to find them:

a=1;b=0;c=0;d=0;e=0;p=0; cycle:n= a*10000+b*1000+c*100+d*10+e;if(n%2==0 and n%3==0 and n%4==0 and n%5==0 and n%6==0 and n%7==0 and n%8==0 and n%9==0, goto loop, goto next); loop:printn,", ",;p=p+1; next:e++;if(e<10, goto cycle, 0);e=0;d++;if(d<10, goto cycle, 0);e=0;d=0;c++;if(c<10, goto cycle,0);c=0;d=0;e=0;b++;if(b<10, goto cycle,0);e=0;d=0;c=0;b=0;a++;if(a<10, goto cycle,0);print"Total = ",p

OUTPUT =( 10080 , 12600 , 15120 , 17640 , 20160 , 22680 , 25200 , 27720 , 30240 , 32760 , 35280 , 37800 , 40320 , 42840 , 45360 , 47880 , 50400 , 52920 , 55440 , 57960 , 60480 , 63000 , 65520 , 68040 , 70560 , 73080 , 75600 , 78120 , 80640 , 83160 , 85680 , 88200 , 90720 , 93240 , 95760 , 98280)> Total =  36

Dec 15, 2019