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If n is a positive integer, which one of the following numbers must have a remainder of 3 when divided by any of the numbers 4, 5, and 6?
- a)12n + 3
- b)24n + 3
- c)90n + 3
- d)120n + 3
Correct answer is option 'D'. Can you explain this answer?
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If n is a positive integer, which one of the following numbers must ha...
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If n is a positive integer, which one of the following numbers must ha...
Solution:
To find the number that leaves a remainder of 3 when divided by 4, 5, and 6, we need to find the LCM of 4, 5, and 6 and add 3 to it.
Step 1: Finding the LCM of 4, 5, and 6.
LCM of 4, 5, and 6 = 2^2 × 3 × 5 = 60
Step 2: Adding 3 to the LCM found in step 1.
60 + 3 = 63
Therefore, any multiple of 63 will leave a remainder of 3 when divided by 4, 5, and 6.
Step 3: Checking the given options.
a) 12n + 3 = 3(4n + 1) - leaves a remainder of 3 when divided by 4 but not when divided by 5 or 6.
b) 24n + 3 = 3(8n + 1) - leaves a remainder of 3 when divided by 3 and 5 but not when divided by 4 or 6.
c) 90n + 3 = 3(30n + 1) - leaves a remainder of 3 when divided by 3 and 6 but not when divided by 4 or 5.
d) 120n + 3 = 3(40n + 1) - leaves a remainder of 3 when divided by 4, 5, and 6.
Therefore, the correct answer is option D, 120n + 3.
To find the number that leaves a remainder of 3 when divided by 4, 5, and 6, we need to find the LCM of 4, 5, and 6 and add 3 to it.
Step 1: Finding the LCM of 4, 5, and 6.
LCM of 4, 5, and 6 = 2^2 × 3 × 5 = 60
Step 2: Adding 3 to the LCM found in step 1.
60 + 3 = 63
Therefore, any multiple of 63 will leave a remainder of 3 when divided by 4, 5, and 6.
Step 3: Checking the given options.
a) 12n + 3 = 3(4n + 1) - leaves a remainder of 3 when divided by 4 but not when divided by 5 or 6.
b) 24n + 3 = 3(8n + 1) - leaves a remainder of 3 when divided by 3 and 5 but not when divided by 4 or 6.
c) 90n + 3 = 3(30n + 1) - leaves a remainder of 3 when divided by 3 and 6 but not when divided by 4 or 5.
d) 120n + 3 = 3(40n + 1) - leaves a remainder of 3 when divided by 4, 5, and 6.
Therefore, the correct answer is option D, 120n + 3.
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If n is a positive integer, which one of the following numbers must ha...
If it should be applicable for all the three 4,5 &6 - it should be a factor of each - hence the LCM of them is 60 ( divisble by 4,5&6) - if this was the case the answer should be 60n + 3 ( remainder) ...the next best option is option D = (60*2)n +3 (remainder is constant)
Or
you can try the trial and error method -
A) 12 - not divisible by 5
B) 24 - not divisble by 5
C) 90 - not divisible by 4
D) 120 - CORRECT Answer as divisble by all:)
Or
you can try the trial and error method -
A) 12 - not divisible by 5
B) 24 - not divisble by 5
C) 90 - not divisible by 4
D) 120 - CORRECT Answer as divisble by all:)
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