Suppose n\mod d = 3. Which of the following is not necessarily true?

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Let's analyze what it means that n mod d = 3 in general terms.
When the remainder is 3, the division algorithm tells us that n = qd + 3 for some integer q ≥ 0, and the remainder 3 must satisfy 0 ≤ 3 < d, which immediately implies d > 3.
Option a: 'n = d + 3'. This would require q = 1 in the expression n = qd + 3. While it is possible for q to be 1, it is not guaranteed; q could be 0, 2, 3, etc. Therefore, this statement is not necessarily true in all cases.
Option b: 'd ∤ n'. The remainder being 3 means n is not divisible by d, since divisibility would require a remainder of 0. Given that the remainder is 3, d cannot divide n. Thus, this statement is necessarily true.
Option c: 'd | (n − 3)'. Subtracting 3 from n gives n − 3 = qd, which is clearly a multiple of d. Hence, d divides n − 3 for all valid n and d in this scenario. This statement is necessarily true.
Option d: 'd > 3'. From the condition that the remainder is 3, the divisor must exceed the remainder, so d must be strictly greater than 3. Therefore, this statement is necessarily true.
In summary, among the options, only statement a does not have to hold in every case where n mod d = 3; the others must be true by the division algorithm and the definition of remainder.

FIT1058_S1_2025 Week 7 Pre-reading Quiz

Suppose n\mod d = 3. Which of the following is not necessarily true?

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