Question

product of first 10 natural numbers 2 power a*3power b*5 power c *7power d then find the value of a+b+c+d​

Answer 1

Answer:

13

Step-by-step explanation:

The natural numbers: 1, 2, 3, 2², 5, 2×3, 7, 2³, 3², 2×5

→ a=1+2+1+3+1=8

→ b=1+1=2

→ c=1+1=2

→ d=1

The answer is a+b+c+d=13.

How To Solve

• Multiplication is an exponent sum.

Answer 2

The value of a+b+c+d​ is 14

• The product of the first 10 natural numbers is:

1 × 2 × 3 × 4 × 5 × 6 × 7 × 8 × 9 × 10 = 3,628,800

• To find the prime factorization of this number, we can start by dividing by 2 repeatedly until we get an odd number:

3,628,800 ÷ 2 = 1,814,400

1,814,400 ÷ 2 = 907,200

907,200 ÷ 2 = 453,600

453,600 ÷ 2 = 226,800

226,800 ÷ 2 = 113,400

113,400 ÷ 2 = 56,700

56,700 ÷ 2 = 28,350

28,350 ÷ 2 = 14,175

14,175 ÷ 2 = 7,087.5 (not divisible by 2)

So the largest power of 2 that divides 3,628,800 is 8 (2 cubed).

• Next, we can divide by 3 repeatedly until we get a number that is not divisible by 3:

7,087.5 ÷ 3 = 2,362.5

2,362.5 ÷ 3 = 787.5

787.5 ÷ 3 = 262.5

262.5 ÷ 3 = 87.5 (not divisible by 3)

So the largest power of 3 that divides 3,628,800 is 4 (3 raised to the power of 4).

• Next, we can divide by 5 repeatedly until we get a number that is not divisible by 5:

87.5 ÷ 5 = 17.5

17.5 ÷ 5 = 3.5 (not divisible by 5)

So the largest power of 5 that divides 3,628,800 is 1 (5 raised to the power of 1).

• Finally, we can divide by 7 repeatedly until we get a number that is not divisible by 7:

3.5 ÷ 7 = 0.5 (not divisible by 7)

So the largest power of 7 that divides 3,628,800 is 1 (7 raised to the power of 1).

Therefore, the prime factorization of 3,628,800 is:

2^8 × 3^4 × 5^1 × 7^1

The sum of the exponents of the prime factors is:

a + b + c + d = 8 + 4 + 1 + 1 = 14

The value of a + b + c + d is 14.

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