On multiplying a certain positive integer by 7, the product is a number in which every digit is 3. What is the smallest such integer for which 7 times the integer consists only of the digit 3?

Introduction:
This question is about number patterns and divisibility. You are told that when a certain integer is multiplied by 7, the result is a number composed entirely of the digit 3 (for example, 33, 333, 3333, and so on). The task is to determine the smallest integer for which this happens. This involves understanding repdigit numbers (numbers with repeated identical digits) and checking divisibility by 7.

Given Data / Assumptions:

• We need a number N such that 7N is a repdigit consisting only of 3s.
• The product 7N may be 33, 333, 3333, 33333, 333333, and so on.
• Among all such valid N, we need the smallest positive integer.
• Answer choices include several candidate integers like 47619 and 476190476.

Concept / Approach:
Repdigit numbers containing only 3s can be expressed as 3 * (111..., a string of 1s). One way to check candidates is to multiply them by 7 and see whether the product has only the digit 3. Since the options are specific and relatively few, direct checking is efficient and reveals the pattern. We must also confirm that there is no smaller candidate with fewer digits that satisfies the condition.

Step-by-Step Solution:
Check 47619: 7 * 47619 = 333333.This product, 333333, clearly consists only of the digit 3.Check 476190476: 7 * 476190476 = 3333333332, which contains a 2 and is therefore not valid.Check 48617: 7 * 48617 = 340319, which has digits other than 3.Check 4587962: 7 * 4587962 = 32115734, again not all 3s.Thus, 47619 is the only candidate among the given options that produces a product consisting entirely of the digit 3.To see that it is smallest, note that 7 * 333 = 2331, 7 * 3333 = 23331, 7 * 33333 = 233331, etc., and none of these produce an all-3 number until we reach 333333 = 7 * 47619.

Verification / Alternative check:
Compute 333333 / 7 = 47619 exactly, confirming no remainder. Also, check smaller repdigits (33, 333, 3333, 33333) to confirm that none are divisible by 7 without leaving other digits in the result. This confirms that 47619 is indeed the smallest integer with the required property.

Why Other Options Are Wrong:
476190476, 48617, and 4587962: their products with 7 contain digits other than 3.33333: this is not the integer N but rather a candidate product; it is not divisible by 7 into a whole repdigit of 3s as required by the problem description.

Common Pitfalls:
Confusing the integer N with the resulting number 7N and trying to match both with repeated 3s.Not actually checking the multiplication and assuming that a long number like 476190476 must be correct.Overlooking that the question asks for the smallest such integer, not any arbitrary integer that might work.

Final Answer:
47619