Question

find the maximum value of N such that 42×57×92×91×52×62×63×64×65×66×67is perfectly divisible by 42​

Answer 1

Given :- find the maximum value of N such that 42×57×92×91×52×62×63×64×65×66×67 is perfectly divisible by 42^N .

A) 4

B) 3

C) 5

D) 6

Solution :-

→ 42×57×92×91×52×62×63×64×65×66×67

→ (6 * 7) * (3 * 19) * (4 * 23) * (7 * 13) * (4 * 13) * (2 * 31) * (3 * 3 * 7) * (2⁶) * (5 * 13) * (6 * 11) * (1 * 67)

taking 6 and 7 common we get,

→ 7³ * 6² * 2 * 3 * (19 * 4 * 23 * 13 * 4 * 13 * 31 * 3 * 3 * 2⁶ * 5 * 13 * 11 * 67)

→ 7³ * 6² * 6 * (19 * 4 * 23 * 13 * 4 * 13 * 31 * 3 * 3 * 2⁶ * 5 * 13 * 11 * 67)

→ 7³ * 6³ * (19 * 4 * 23 * 13 * 4 * 13 * 31 * 3 * 3 * 2⁶ * 5 * 13 * 11 * 67)

→ (7 * 6)³ * (19 * 4 * 23 * 13 * 4 * 13 * 31 * 3 * 3 * 2⁶ * 5 * 13 * 11 * 67)

→ (42)³ * (19 * 4 * 23 * 13 * 4 * 13 * 31 * 3 * 3 * 2⁶ * 5 * 13 * 11 * 67)

therefore,

→ 42³ = 42^N

→ N = 3 (B) (Ans.)

Hence, Maximum value of N such the given expression is divisible by 42 will be 3 .

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