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If n is a natural number, then 92n - 42n is always divisible by ____.
- a)5
- b)13
- c)both (a) and (b)
- d)neither (a) nor (b)
Correct answer is option 'C'. Can you explain this answer?
Most Upvoted Answer
If n is a natural number, then 92n - 42n is always divisible by ____.a...
Given expression is in the form a2 - b2
92n - 42n is divisible by both (9 - 4) and (9 + 4) i.e. 5 and 13.
92n - 42n is divisible by both (9 - 4) and (9 + 4) i.e. 5 and 13.
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If n is a natural number, then 92n - 42n is always divisible by ____.a...
To determine whether 92n - 42n is always divisible by 5, 13, or both, we need to consider the properties of divisibility for each number.
Divisibility by 5:
A number is divisible by 5 if its units digit is either 0 or 5. Let's examine the expression 92n - 42n:
92n - 42n = 90n + 2n - 40n + 2n
= 92n - 40n + 4n
= 52n + 4n
= 56n
Since the units digit of 56n is always 6, it is not divisible by 5. Therefore, 92n - 42n is not always divisible by 5.
Divisibility by 13:
To determine divisibility by 13, we can use the rule that a number is divisible by 13 if the difference between three times its units digit and the remaining digits is divisible by 13. Let's apply this rule to 92n - 42n:
92n - 42n = 90n + 2n - 40n + 2n
= 92n - 40n + 4n
= 52n + 4n
= 56n
We can see that 56n follows the pattern for divisibility by 13. For example, when n = 1, 56n = 56, and the difference between three times 6 (18) and the remaining digit 5 is 13, which is divisible by 13. This pattern holds true for any natural number n.
Therefore, we can conclude that 92n - 42n is always divisible by 13.
Combining the results:
Since 92n - 42n is not always divisible by 5 but is always divisible by 13, the correct answer is both (a) and (b), option C.
Divisibility by 5:
A number is divisible by 5 if its units digit is either 0 or 5. Let's examine the expression 92n - 42n:
92n - 42n = 90n + 2n - 40n + 2n
= 92n - 40n + 4n
= 52n + 4n
= 56n
Since the units digit of 56n is always 6, it is not divisible by 5. Therefore, 92n - 42n is not always divisible by 5.
Divisibility by 13:
To determine divisibility by 13, we can use the rule that a number is divisible by 13 if the difference between three times its units digit and the remaining digits is divisible by 13. Let's apply this rule to 92n - 42n:
92n - 42n = 90n + 2n - 40n + 2n
= 92n - 40n + 4n
= 52n + 4n
= 56n
We can see that 56n follows the pattern for divisibility by 13. For example, when n = 1, 56n = 56, and the difference between three times 6 (18) and the remaining digit 5 is 13, which is divisible by 13. This pattern holds true for any natural number n.
Therefore, we can conclude that 92n - 42n is always divisible by 13.
Combining the results:
Since 92n - 42n is not always divisible by 5 but is always divisible by 13, the correct answer is both (a) and (b), option C.
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