Consider all 3-digit numbers (without repetition of digits) obtained using three non-zero digits which are…
Consider all 3-digit numbers (without repetition of digits) obtained using three non-zero digits which are multiples of 3. Let S be their sum.
Which of the following is/are correct?
1. S is always divisible by 74.
2. S is always divisible by 9.
Select the correct answer using the code given below:
1 only
2 only
Both 1 and 2
Neither 1 nor 2
Solution
Three non-zero digits which are multiples of 3 are 3, 6 and 9. Using these we form $3! = 6$ three-digit numbers. Their sum $S = 369 + 396 + 639 + 693 + 936 + 963 = 3996$. Since $3996 = 74 \times 54 = 9 \times 444$, S is divisible by both 74 and 9. Hence both statements are correct.