Question 8 of 37advanced🔧 ApplyNumerical3 marks

Find the smallest number that is a multiple of all the numbers from 1 to 10.

Correct Answer

Given: Find the smallest number that is a multiple of all numbers from 1 to 10.

Method: Find LCM of 1, 2, 3, 4, 5, 6, 7, 8, 9, 10.

Step 1: Prime factorizations: 2 = 2 3 = 3 4 = 2² 5 = 5 6 = 2×32 \times 3 7 = 7 8 = 2³ 9 = 3² 10 = 2×52 \times 5

Step 2: Take highest powers of each prime factor: 2³, 3², 5, 7

Step 3: Calculate LCM: LCM = 2³ × 3² × 5×75 \times 7 = 8×98 \times 9 × 5×75 \times 7 = 2520

Answer: The smallest number is 2520.

Exercise: Figure it Out (Section 2) | Q: 10 | (Chapter: Page 5)
For More Understanding

Explanation

We need LCM of 1,2,3,4,5,6,7,8,9,10. Prime factorizations: 2=2, 3=3, 4=2², 5=5, 6=2×32 \times 3, 7=7, 8=2³, 9=3², 10=2×52 \times 5. LCM = 2³ × 3² × 5×75 \times 7 = 8×98 \times 9 × 5×75 \times 7 = 2520.

Solution Steps

  1. Step 1: Find LCM of 1,2,3,4,5,6,7,8,9,10.

  2. Step 2: Prime factorizations: 2=2, 3=3, 4=2², 5=5, 6=2×32 \times 3, 7=7, 8=2³, 9=3², 10=2×52 \times 5.

  3. Step 3: Take highest powers: 2³, 3², 5, 7.

  4. Step 4: LCM = 2³ × 3² × 5×75 \times 7 = 8×98 \times 9 × 5×75 \times 7 = 2520.

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