If two integers are each chosen at random from the first 50

Question

If two integers are each chosen at random from the first 50 positive integers, what is the probability that their product or their sum is divisible by 5?

(A)  12/25 
(B)  1/2 
(C)  13/25 
(D)  14/25 
(E)  16/25

sarkarsarbani · Answer

can someone post the soln?

kevincan · Answer

It’s easier to find the probability that neither the product nor the sum is a multiple of 5.

Suppose that x is an integer from 1 to 50 chosen at random from the first 50 positive integers and that r is an integer from 0 to 4. The probability that x yields a remainder of r when it is divided by 5 is 1/5. Why? Because there are five equals likely remainders : 0,1,2,3,4.
 
For neither the sum nor the product to be a multiple of 5, the remainder r when the first integer is divided by 5 must be different from 0 (leaving us with 4 of the 5 possibilities) and the remainder when the second integer is divided by 5 must be different from both 0 and 5 - r (leaving us 3 of the 5 possibilities).

Thus the probability that neither the product nor the sum of the two integers is a multiple of 5 is 4/5 * 3/5 = 12/25

Therefore , the required probability is 1 -12/25 = 13/25

Flyingdutch1307 · Answer

Instead of counting actual numbers, use remainder probability directly.
In 1-50, each remainder (0,1,2,3,4) appears exactly 10 times out of 50. So probability of any remainder = 1/5. That's the shortcut that bypasses heavy P&C Calcuations entirely.
The logic then becomes:

Case 1 (Product Not Divisible): remainder ≠ 0 → probability 4/5
Case 2 (Sum Not Divisible): remainder ≠ 0 AND remainder ≠ (5-r) to avoid sum divisible by 5 || out of these 4 numbers (1,2,3,4) only 3 can be possible, For Ex: if you pick your 1st Integer which has remainder 2, then only 1,4 is possible as remainder for #2 Integer... Hence, only 3 possible cases... so → probability 3/5

Combined Probability (Product & Sum Not Divisible ) => 
P(AB) = P(A)*P(B)

P(!AB) = 1 - P(AB)
=>
= 1 - 4/5 * 3/5
= 13/25

If two integers are each chosen at random from the first 50

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If two integers are each chosen at random from the first 50

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

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