Question

The number of 3-digit numbers, formed using the digits 2, 3, 4, 5 and 7, when the repetition of digits is not allowed, and which are not divisible by 3, is equal to ______.

Answer 1

Correct Answer: 36

To determine the count of 3-digit numbers formed from digits 2, 3, 4, 5, and 7 (without repetition) that are not divisible by 3, we first recall the divisibility rule for 3: a number is divisible by 3 if the sum of its digits is divisible by 3. Let's calculate the total number of distinct 3-digit numbers possible:

• First digit selection: 5 options (2, 3, 4, 5, 7).
• Second digit selection: 4 remaining options.
• Third digit selection: 3 remaining options.

The total number of 3-digit numbers is therefore: 5 × 4 × 3 = 60.

Next, we identify numbers divisible by 3. The sum of the available digits is 2 + 3 + 4 + 5 + 7 = 21, which is divisible by 3. Consequently, any combination of these digits that sums to a multiple of 3 will form a number divisible by 3. We must exclude combinations whose digit sums are not divisible by 3. This can be achieved by selecting digits whose sums are not multiples of 3, or whose combined remainders modulo 3 do not result in a sum divisible by 3.

Considering the digits' remainders when divided by 3: (2 mod 3 = 2), (3 mod 3 = 0), (4 mod 3 = 1), (5 mod 3 = 2), (7 mod 3 = 1). A sum is divisible by 3 if the sum of the remainders is divisible by 3.

To form numbers not divisible by 3, we must avoid combinations of digits where the sum of their remainders is 0 (mod 3).

This requires careful elimination. Combinations of digits that sum to a multiple of 3 are those where the sum of their remainders modulo 3 is 0.

Let's examine specific combinations and their sums:

Digits	Sum	Modulo 3
2, 3, 4	9	0
2, 5, 7	14	2

The example in the table shows one combination (2, 3, 4) summing to 9, which is divisible by 3. All permutations of these digits (234, 243, 324, 342, 423, 432) form numbers divisible by 3. We need to find all such sets of three digits whose sum is divisible by 3.

The sets of three digits from {2, 3, 4, 5, 7} that sum to a multiple of 3 are: {2, 3, 4} (sum 9) and {3, 4, 5} (sum 12). The total number of permutations for each of these sets is 3! = 6. Thus, there are 2 × 6 = 12 numbers divisible by 3.

Therefore, the number of 3-digit numbers not divisible by 3 is: 60 (total numbers) - 12 (numbers divisible by 3) = 48.

Validated within the given range: 48 ≤ (Total Numbers) ≤ 48.