{"id":"aqua_rat:109d7772b2baf19569fe48916912f3b1","source_id":"aqua_rat","source_item_id":"109d7772b2baf19569fe48916912f3b1","external_id":null,"program":null,"section":"math","category_code":"PHM","category":"Preparing for Higher Math","subcategory_code":null,"subcategory":null,"form_id":null,"passage_id":null,"position":null,"scored":true,"layout_warning":null,"stem":"If Q, a positive integer, has 5 factors, which of the following must be true about Q? I. Q is the square of a prime number. II. Q is the fourth power of a prime number. III. Q is the product of two prime numbers.","stimulus":"","stimulus_html":null,"stem_html":"<p>If Q, a positive integer, has 5 factors, which of the following must be true about Q?</p><p>I. Q is the square of a prime number.</p><p>II. Q is the fourth power of a prime number.</p><p>III. Q is the product of two prime numbers.</p>","rationale_html":"<p>If Q has 5 factors, we can represent Q = a^4, where a is positive integer more than 1.Let&#x27;s assume that &quot;a&quot; is not a prime number. Let a = kp, where both k and p are positive integers.</p><p>Thus, Q = (kp)4=k4∗p4(kp)4=k4∗p4. Now the number of factors of Q = (4+1)*(4+1) = 25. But as the given condition states that Q has ONLY 5 factors, thus &quot;a&quot; can&#x27;t have any other factor except 1 and itself. Thus, a = prime number.</p><p>Statement I :We can represent Q = (a^2)^2. Thus, we have to prove whether a^2 is a prime number. Take a=2. We can see that it is not a prime number. Thus, this option can&#x27;t answer a &quot;MUST be true question&quot;</p><p>Statement II : Always true as proved above.</p><p>Statement III : Again take a =2. Thus, Q = 64. We don&#x27;t have this as product of 2 primes.</p><p>The Answer is, B.</p>","correct_answer":["B"],"equiv_key":"be1254e21a3c622bcb4e48c0","has_media":false,"has_mathml":false,"has_table":false,"section_label":"Mathematics","options":[{"label":"A","content_html":"<p>I only</p>","ord":0},{"label":"B","content_html":"<p>III only</p>","ord":1},{"label":"C","content_html":"<p>II only</p>","ord":2},{"label":"D","content_html":"<p>I and II only</p>","ord":3},{"label":"E","content_html":"<p>I and III only</p>","ord":4}],"passage":null,"attribution":{"source_id":"aqua_rat","source_name":"AQuA-RAT algebra word problems with rationales","rights_holder":"Google DeepMind and the AQuA-RAT contributors","canonical_url":"https://github.com/google-deepmind/AQuA","retrieved_from":"https://raw.githubusercontent.com/google-deepmind/AQuA/master/","attribution":"Algebra word problems from the AQuA-RAT dataset (Ling et al., 2017), published by Google DeepMind under the Apache License 2.0.","disclaimer":"AQuA-RAT items are crowdsourced algebra problems, not ACT questions. They carry five answer choices in the pre-2025 ACT Mathematics style and no ACT reporting category. Useful for drilling Preparing for Higher Math content; not a substitute for a real form.","license":"Apache License 2.0. Crowdsourced by DeepMind and published for free use, including commercially.","redistributable":true,"item_count":508},"equivalents":[]}