lean_workbook_plus_61447

⚠️ Retired — specification defect

The Lean statement below does not encode the problem shown on this page, so its Disproved status carries no information about that problem. Do not import this node or use it as a dependency.

If $n$ leaves a remainder of $4,5,$ or $6$ when divided by $7$ , it will be rounded up. If $n$ is divisible by $7$ , the above equality is trivial. There are $\lfloor \frac{100}{7} \rfloor = 14$ groups of $7$ between $1$ and $100$ . In each group, there are 4 values of $n$ that work, giving $14 \cdot 4 = 56$ solutions. We have not considered numbers past $14 \cdot 7 = 98$ , of which there is only one, $99$ , which does not work. Therefore, the answer is $\boxed{56}$

Why this node was retired

The posted statement is

theorem lean_workbook_plus_61447 :
  Finset.card (Finset.filter (λ n => 1 ≤ n ∧ n ≤ 100 ∧ n % 7 = 4 ∨ n % 7 = 5 ∨ n % 7 = 6) (Finset.Icc 1 100)) = 56   :=  by sorry

The source counted residue0 as a fourth case, but the formal filter only contains4,5,6.

The recorded counterexample refutes the statement as encoded. It says nothing about the problem shown above, which is a different proposition.

Proposed corrected statement

Count n∈{1,…,100} with n mod7∈{0,4,5,6}; the count is56. With the posted three residues the count is42.

Diagnosis and correction from the public Prove2Me statement audit (wamlat/prove2me-errors). The correction is natural-language mathematics and is not Lean-verified — it is a specification for a corrected node, not a drop-in replacement. No corrected replacement node exists yet.