Solve this: 19 Consider a three digit number x1x2x3 such that x1 x2 x3 ∈ N Then the - Maths - Sequences and Series

Question

Solve this:
19 Consider a three digit number
x1x2x3 such that x1
x2 x3 ∈ N Then the number of positive
integral solutions of x1 x2 x3 =
480 is λ 2 -100, then the sum of the digits of λ is
(A) 8 (B) 9 (C) 7 (D) 10

Shivam Mishra · Accepted Answer

Dear Student,
Please find below the solution to the asked query:

We havex1x2x3=480=⇒x1x2x3=25×31×51Let x1=2α1
3β1 5γ1x2=2α2 3β2 5γ2x3=2α3 3β3
5γ3⇒2α1 3β1 5γ12α2 3β2
5γ22α3 3β3
5γ3=25×31×51⇒2α1+α2+α3
3β1+β2+β3
5γ1+γ2+γ3=25×31×51⇒α1+α2+α3=5β1+β2+β3=1γ1+γ2+γ3=1Now we know that number of non-negative integral solutions of x1+x2+
+xr=n is n+r-1Cr-1HenceNumber of non-negative integral solutions of α1+α2+α3=5 is 5+2C2=7C2Number of non-negative integral solutions of β1+β2+β3=1 is 1+2C2=3C2Number of non-negative integral solutions of γ1+γ2+γ3=1 is 1+2C2=3C2⇒λ2-100=7C2×3C2×3C2=7×62×13×22×13×22×1⇒λ2-100=189⇒λ2=289⇒λ=289=17Sum of digits=1+7=8

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Solve this: 19. Consider a three digit number x1x2x3 such that x1.x2.x3 ∈ N. Then the number of positive integral solutions of x1.x2.x3 = 480 is λ 2 -100, then the sum of the digits of λ is (A) 8 (B) 9 (C) 7 (D) 10

Solve this:
19. Consider a three digit number x₁x₂x₃ such that x₁.x₂.x₃ $\in$ N. Then the number of positive integral solutions of x₁.x₂.x₃= 480 is $λ^{2}$ -100, then the sum of the digits of $|λ|$ is
(A) 8 (B) 9 (C) 7 (D) 10