Question: Suppose $ p $ is a positive multiple of 7. If $ p $ squared is less than 1000, what is the largest possible value of $ p $?
["Question: Suppose $ p $ is a positive multiple of 7. If $ p $ squared is less than 1000, what is the largest possible value of $ p $?", "When tackling math problems involving multiples and inequalities, clarifying key constraints helps narrow down possible answers. In this case, we are given two important conditions:", "- $ p $ is a positive multiple of 7, meaning $ p = 7k $ for some positive integer $ k $.\n- $ p^2 < 1000 $.", "To find the largest possible value of $ p $ satisfying both conditions, follow these steps:", "### Step 1: Use the inequality\nStart by solving the inequality:\n$$\np^2 < 1000\n$$\nTake the square root of both sides:\n$$\np < \sqrt{1000} \approx 31.62\n$$\nSince $ p $ must be a positive multiple of 7, we seek the largest multiple of 7 less than 31.62.", "### Step 2: List multiples of 7 below 31.62\nThe positive multiples of 7 are:\n$$\n7, 14, 21, 28, 35, \dots\n$$\nCheck which of these satisfy $ p^2 < 1000 $:\n- $ 7^2 = 49 < 1000 $ ✔️\n- $ 14^2 = 196 < 1000 $ ✔️\n- $ 21^2 = 441 < 1000 $ ✔️\n- $ 28^2 = 784 < 1000 $ ✔️\n- $ 35^2 = 1225 > 1000 $ ❌ (too large)", "The largest valid multiple is $ p = 28 $, since $ 28^2 = 784 < 1000 $, but $ 35 $ exceeds the limit.", "### Step 3: Confirm no larger multiple of 7 fits\nThe next multiple, 35, is already over 31.62, so no smaller valid multiple of 7 gives a larger $ p $.", "---", "Conclusion:\nThe largest positive multiple of 7 such that $ p^2 < 1000 $ is\n$$\n\boxed{28}\n$$", "This problem highlights how constraints on multiples and inequalities together allow efficient identification of the maximum valid value—ideal for learners mastering basic number theory and inequality solving."]
