Middle School Math Quiz: Gcf And Lcm
3 questions · exam conditions
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Gcf And LcmQuestion 1 of 3

A number nn is divisible by both 15 and 21. What is the smallest possible value of nn that is greater than 200?

210
315
420
105
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Middle School Math Quiz

Middle School Math Quiz: Gcf And Lcm

Practice Gcf And Lcm in Middle School Math with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Gcf And Lcm, giving you a quick way to practice the rules, question types, and explanations that matter most for Middle School Math.

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Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

A number nn is divisible by both 15 and 21. What is the smallest possible value of nn that is greater than 200?

  1. 210 (correct answer)
  2. 315
  3. 420
  4. 105
Explanation: If nn is divisible by both 15 and 21, then nn must be divisible by LCM(15,21). Find the LCM: 15 = 3×5 and 21 = 3×7, so LCM = 3×5×7 = 105. Therefore nn must be a multiple of 105. The multiples of 105 are: 105, 210, 315, 420, ... Since we need n>200n > 200, the smallest such multiple is 210. We can verify: 210 ÷ 15 = 14 and 210 ÷ 21 = 10, so 210 is indeed divisible by both numbers.

Question 2

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?

  1. 5 rotations (correct answer)
  2. 3 rotations
  3. 15 rotations
  4. 25 rotations
Explanation: 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.

Question 3

The GCF of two numbers is 8, and their LCM is 240. If one of the numbers is 48, what is the other number?

  1. 30
  2. 40 (correct answer)
  3. 32
  4. 24
Explanation: 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.