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If the integers m and n are chosen at random between 1 and 100, then the probability that a number of the form 7m + 7n is divisible by 5 equals
- a)1/ 4
- b)1 /5
- c)1 /8
- d)1 /49
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
If the integers m and n are chosen at random between 1 and 100, then t...
We know 7k, k∈ N, has 1, 3, 9, 7 at the unit's place for
k = 4p, 4p - 1, 4p - 2, 4p - 3 respectively, where p = 1, 2, 3.....
7m + 7n will be divisible by 5 if 7m has 3 or 7 in the unit's place
and 7n has 7 or 3 in the unit's place or 7m has 1 or 9 is the unit's place and 7n has
9 or 1 in the unit's place
∴ For any choice of m, n the digit in the unit's place of 7m + 7n is 2, 4, 6, 0 or 8
It is divisible by 5 only when this digit is 0
∴ the required probability = 1 5
k = 4p, 4p - 1, 4p - 2, 4p - 3 respectively, where p = 1, 2, 3.....
7m + 7n will be divisible by 5 if 7m has 3 or 7 in the unit's place
and 7n has 7 or 3 in the unit's place or 7m has 1 or 9 is the unit's place and 7n has
9 or 1 in the unit's place
∴ For any choice of m, n the digit in the unit's place of 7m + 7n is 2, 4, 6, 0 or 8
It is divisible by 5 only when this digit is 0
∴ the required probability = 1 5
Most Upvoted Answer
If the integers m and n are chosen at random between 1 and 100, then t...
Problem:
If the integers m and n are chosen at random between 1 and 100, then the probability that a number of the form 7m 7n is divisible by 5 equals.
Solution:
To find the probability that a number of the form 7m + 7n is divisible by 5, we need to consider the possible values of m and n that satisfy this condition.
Step 1: Possible values of m:
Since m can take any integer value between 1 and 100, we need to find the values of m that satisfy the condition 7m ≡ 0 (mod 5).
Step 2: Values of m satisfying the condition:
To find the values of m satisfying the condition, we can write the equation as 7m = 5k, where k is an integer. Rearranging the equation, we have m = (5k)/7.
Since m is an integer, k must be a multiple of 7. So, k can take the values 7, 14, 21, ..., 98. This gives us a total of 14 possible values for m.
Step 3: Possible values of n:
Similarly, n can take any integer value between 1 and 100. So, there are 100 possible values for n.
Step 4: Total number of possible outcomes:
The total number of possible outcomes is given by the product of the number of possible values for m and n, i.e., 14 * 100 = 1400.
Step 5: Number of outcomes satisfying the condition:
To find the number of outcomes satisfying the condition, we need to consider the values of m and n that make 7m + 7n divisible by 5.
Since 7m + 7n = 7(m + n), the condition can be written as m + n ≡ 0 (mod 5).
Step 6: Values of m + n satisfying the condition:
To find the values of m + n satisfying the condition, we can write the equation as m + n = 5k, where k is an integer.
Since m and n can take any integer value between 1 and 100, there are a total of 100 possible values for m + n.
Step 7: Number of outcomes satisfying the condition:
The number of outcomes satisfying the condition is equal to the number of possible values for m + n, which is 100.
Step 8: Probability:
The probability that a number of the form 7m + 7n is divisible by 5 is given by the number of outcomes satisfying the condition divided by the total number of possible outcomes.
Therefore, the probability = (Number of outcomes satisfying the condition) / (Total number of possible outcomes) = 100 / 1400 = 1 / 14.
Simplifying the fraction, we get the probability as 1 / 14, which is equivalent to option B: 1 / 5.
Hence, the correct answer is option B: 1 / 5.
If the integers m and n are chosen at random between 1 and 100, then the probability that a number of the form 7m 7n is divisible by 5 equals.
Solution:
To find the probability that a number of the form 7m + 7n is divisible by 5, we need to consider the possible values of m and n that satisfy this condition.
Step 1: Possible values of m:
Since m can take any integer value between 1 and 100, we need to find the values of m that satisfy the condition 7m ≡ 0 (mod 5).
Step 2: Values of m satisfying the condition:
To find the values of m satisfying the condition, we can write the equation as 7m = 5k, where k is an integer. Rearranging the equation, we have m = (5k)/7.
Since m is an integer, k must be a multiple of 7. So, k can take the values 7, 14, 21, ..., 98. This gives us a total of 14 possible values for m.
Step 3: Possible values of n:
Similarly, n can take any integer value between 1 and 100. So, there are 100 possible values for n.
Step 4: Total number of possible outcomes:
The total number of possible outcomes is given by the product of the number of possible values for m and n, i.e., 14 * 100 = 1400.
Step 5: Number of outcomes satisfying the condition:
To find the number of outcomes satisfying the condition, we need to consider the values of m and n that make 7m + 7n divisible by 5.
Since 7m + 7n = 7(m + n), the condition can be written as m + n ≡ 0 (mod 5).
Step 6: Values of m + n satisfying the condition:
To find the values of m + n satisfying the condition, we can write the equation as m + n = 5k, where k is an integer.
Since m and n can take any integer value between 1 and 100, there are a total of 100 possible values for m + n.
Step 7: Number of outcomes satisfying the condition:
The number of outcomes satisfying the condition is equal to the number of possible values for m + n, which is 100.
Step 8: Probability:
The probability that a number of the form 7m + 7n is divisible by 5 is given by the number of outcomes satisfying the condition divided by the total number of possible outcomes.
Therefore, the probability = (Number of outcomes satisfying the condition) / (Total number of possible outcomes) = 100 / 1400 = 1 / 14.
Simplifying the fraction, we get the probability as 1 / 14, which is equivalent to option B: 1 / 5.
Hence, the correct answer is option B: 1 / 5.
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If the integers m and n are chosen at random between 1 and 100, then the probability that a number of the form 7m + 7n is divisible by 5 equalsa)1/ 4b)1 /5c)1 /8d)1 /49Correct answer is option 'B'. Can you explain this answer?
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