Question

Form all possible 3 digit numbers using all the digits 2,3 and 4 and find their sum. Check whether the sum is divisible by 37! Is it true for the sum of all the numbers formed by the digits a,b and c of the number abc?

Answer 1

the all possible 3 digit numbers are

243 + 234 + 324 + 342 + 432 + 423 = 1998/37 = 54

yes the sum of all possible 3 digit numbers is divisible by 37

checking whether it is true for all numbers :

abc = 100a + 10b + c

acb = 100a + 10c + b

bca = 100b + 10c + a

bac = 100b + 10a + c

cab = 100c + 10a + b

cba = 100c + 10b + a

abc + acb + bca + bac + cab + cba = 222a + 222b + 222c

abc + acb + bca + bac + cab + cba = 222(a + b + c)

abc + acb + bca + bac + cab + cba = (37 × 6)(a + b + c)

we can see that sum of all the numbers formed by the digits a,b,c of number abc is always divisible by 37.

so it is true that the sum of all the numbers formed by the digits a,b,c of number abc is divisible by 37.