GCF & LCM - Pre Algebra
- The GCF of two numbers is 8, and their LCM is 240. If one of the numbers is 48, what is the other number? Answer: 40 Use the fundamental relationship: GCF(a,b) × LCM(a,b) = a × b. Given: GCF = 8, LCM = 240, one number is 48. So 8 × 240 = 48 × b, where b is the unknown number. This gives us 1920 = 48b, so b = 1920 ÷ 48 = 40. Let's verify: GCF(48,40): 48 = 2⁴×3, 40 = 2³×5. GCF = 2³ = 8 ✓. LCM(48,40): LCM = 2⁴×3×5 = 16×15 = 240 ✓. Choice A (30) would give GCF(48,30) = 6, not 8. Choice C (32) would give GCF(48,32) = 16, not 8. Choice D (24) would give GCF(48,24) = 24, not 8.
- Two gears have 45 and 75 teeth respectively. If they start aligned with specific marked teeth touching, how many complete rotations will the smaller gear make before the marked teeth touch again? Answer: 5 rotations The marked teeth will touch again when both gears return to their starting position. This happens when each gear completes a whole number of rotations, which occurs after LCM(45,75) teeth have passed the contact point. Find LCM(45,75): 45 = 3²×5, 75 = 3×5². LCM = 3²×5² = 9×25 = 225. So after 225 teeth pass, the marked teeth align again. The smaller gear (45 teeth) will have made 225 ÷ 45 = 5 complete rotations. The larger gear (75 teeth) will have made 225 ÷ 75 = 3 rotations. Choice B (3) is the number of rotations the larger gear makes. Choice C (15) might come from incorrectly using GCF instead of LCM. Choice D (25) doesn't correspond to the correct calculation.
- A number is divisible by both 15 and 21. What is the smallest possible value of that is greater than 200? Answer: 210 If is divisible by both 15 and 21, then must be divisible by LCM(15,21). Find the LCM: 15 = 3×5 and 21 = 3×7, so LCM = 3×5×7 = 105. Therefore must be a multiple of 105. The multiples of 105 are: 105, 210, 315, 420, ... Since we need , the smallest such multiple is 210. We can verify: 210 ÷ 15 = 14 and 210 ÷ 21 = 10, so 210 is indeed divisible by both numbers.