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If two integers m and n are chosen at random from 1 to 100 then the probability that a number of the form 7m + 7n is divisible by 5 equals:
Correct answer is '0.25'. Can you explain this answer?
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Verified Answer
If two integers m and n are chosen at random from 1 to 100 then the pr...
Since m and n are selected between 1 and 100, hence sample space = 100 × 100
Also, 71 = 7, 72 = 49, 73 = 343, 74 = 2401,
75 = 16807 etc. Hence 1, 3, 7 and 9 will be the last digits in the powers of 7.
n, m →
↓
1,1 1,2 1,3 ........... 1, 100
2,1 2,2 2,3 ........... 2, 100
...........................................
100, 1 100,2 100, 3......... 100, 100
For m = 1 ; n = 3, 7, 11, ...... 97
∴ Favourable cases = 25
For m = 2, n = 4, 8, 12, ......., 100
∴ Favourable cases = 25
Similarly for every m, favourable n are 25.
∴ Total favourable cases = 100 × 25
Hence required probability =
Also, 71 = 7, 72 = 49, 73 = 343, 74 = 2401,
75 = 16807 etc. Hence 1, 3, 7 and 9 will be the last digits in the powers of 7.
n, m →
↓
1,1 1,2 1,3 ........... 1, 100
2,1 2,2 2,3 ........... 2, 100
...........................................
100, 1 100,2 100, 3......... 100, 100
For m = 1 ; n = 3, 7, 11, ...... 97
∴ Favourable cases = 25
For m = 2, n = 4, 8, 12, ......., 100
∴ Favourable cases = 25
Similarly for every m, favourable n are 25.
∴ Total favourable cases = 100 × 25
Hence required probability =
Most Upvoted Answer
If two integers m and n are chosen at random from 1 to 100 then the pr...
Explanation:
To find the probability that a number of the form 7m + 7n is divisible by 5, we need to consider the remainders of 7m and 7n when divided by 5.
Similarly, we can find the possible remainders of 7n when divided by 5.
Now, we need to find the pairs of remainders (r1, r2) such that r1 + r2 is divisible by 5. There are only four such pairs: (1, 4), (2, 3), (3, 2), and (4, 1).
The probability of choosing a pair of remainders that add up to one of these four pairs is 4/25. Since there are 100 choices for m and 100 choices for n, the total number of pairs (m, n) is 100 * 100 = 10,000.
Therefore, the probability of choosing a pair of integers (m, n) such that 7m + 7n is divisible by 5 is:
P = (number of pairs (m, n) such that 7m + 7n is divisible by 5) / (total number of pairs (m, n))
P = 4/25 * 1/100 * 1/100 = 0.00016
So the probability that a number of the form 7m + 7n is divisible by 5 is 0.00016, which is equivalent to 0.25%.
To find the probability that a number of the form 7m + 7n is divisible by 5, we need to consider the remainders of 7m and 7n when divided by 5.
- If m is 1, then the possible remainders of 7m when divided by 5 are 2 and 4.
- If m is 2, then the possible remainders of 7m when divided by 5 are 4 and 3.
- If m is 3, then the possible remainders of 7m when divided by 5 are 1 and 2.
- If m is 4, then the possible remainders of 7m when divided by 5 are 3 and 1.
- If m is 5, then the possible remainders of 7m when divided by 5 are 0 and 4.
- If m is 6, then the possible remainders of 7m when divided by 5 are 2 and 3.
- If m is 7, then the possible remainders of 7m when divided by 5 are 4 and 1.
- If m is 8, then the possible remainders of 7m when divided by 5 are 1 and 3.
- If m is 9, then the possible remainders of 7m when divided by 5 are 3 and 2.
- If m is 10, then the possible remainders of 7m when divided by 5 are 0 and 1.
Similarly, we can find the possible remainders of 7n when divided by 5.
Now, we need to find the pairs of remainders (r1, r2) such that r1 + r2 is divisible by 5. There are only four such pairs: (1, 4), (2, 3), (3, 2), and (4, 1).
The probability of choosing a pair of remainders that add up to one of these four pairs is 4/25. Since there are 100 choices for m and 100 choices for n, the total number of pairs (m, n) is 100 * 100 = 10,000.
Therefore, the probability of choosing a pair of integers (m, n) such that 7m + 7n is divisible by 5 is:
P = (number of pairs (m, n) such that 7m + 7n is divisible by 5) / (total number of pairs (m, n))
P = 4/25 * 1/100 * 1/100 = 0.00016
So the probability that a number of the form 7m + 7n is divisible by 5 is 0.00016, which is equivalent to 0.25%.
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If two integers m and n are chosen at random from 1 to 100 then the probability that a number of the form 7m+ 7nis divisible by 5 equals:Correct answer is '0.25'. Can you explain this answer?
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If two integers m and n are chosen at random from 1 to 100 then the probability that a number of the form 7m+ 7nis divisible by 5 equals:Correct answer is '0.25'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about If two integers m and n are chosen at random from 1 to 100 then the probability that a number of the form 7m+ 7nis divisible by 5 equals:Correct answer is '0.25'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If two integers m and n are chosen at random from 1 to 100 then the probability that a number of the form 7m+ 7nis divisible by 5 equals:Correct answer is '0.25'. Can you explain this answer?.
If two integers m and n are chosen at random from 1 to 100 then the probability that a number of the form 7m+ 7nis divisible by 5 equals:Correct answer is '0.25'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about If two integers m and n are chosen at random from 1 to 100 then the probability that a number of the form 7m+ 7nis divisible by 5 equals:Correct answer is '0.25'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If two integers m and n are chosen at random from 1 to 100 then the probability that a number of the form 7m+ 7nis divisible by 5 equals:Correct answer is '0.25'. Can you explain this answer?.
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