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n = 234yzn
is a positive integer whose tens and units digits are y and z respectively. It is given that n is divisible by 4, 5 and 9. Find n.
- a)23400
- b)23401
- c)23410
- d)23411
- e)23412
Correct answer is option 'A'. Can you explain this answer?
Most Upvoted Answer
n = 234yznis a positive integer whose tens and units digits are y and ...
Given, n = 234yzn
- The number is divisible by 4 if the last two digits of n are divisible by 4.
- The number is divisible by 5 if the last digit of n is either 0 or 5.
- The number is divisible by 9 if the sum of the digits of n is divisible by 9.
Using the above information, we can deduce the following:
- Since n is divisible by 5, the last digit of n must be 0 or 5.
- Since n is divisible by 9, the sum of its digits is divisible by 9.
- Since y and z are the tens and units digits of n respectively, we can write n as:
n = 23400 + 10y + z
Now, we can use the divisibility rule for 9 to get:
2 + 3 + 4 + 0 + 0 + y + z = 9k
9 + y + z = 9k
y + z = 9(k-1)
Since y and z are digits, their sum can be at most 18. Therefore, k must be 2, which gives:
y + z = 9
We also know that n is divisible by 4, so the last two digits of n must be divisible by 4. This means that the number formed by yz must be divisible by 4. The possible pairs of digits that satisfy this condition and y + z = 9 are:
45, 81
Out of these, only 45 satisfies n = 23400 + 10y + z. Therefore, we have:
n = 23400 + 45
n = 23445
Hence, the correct answer is option A (23400).
- The number is divisible by 4 if the last two digits of n are divisible by 4.
- The number is divisible by 5 if the last digit of n is either 0 or 5.
- The number is divisible by 9 if the sum of the digits of n is divisible by 9.
Using the above information, we can deduce the following:
- Since n is divisible by 5, the last digit of n must be 0 or 5.
- Since n is divisible by 9, the sum of its digits is divisible by 9.
- Since y and z are the tens and units digits of n respectively, we can write n as:
n = 23400 + 10y + z
Now, we can use the divisibility rule for 9 to get:
2 + 3 + 4 + 0 + 0 + y + z = 9k
9 + y + z = 9k
y + z = 9(k-1)
Since y and z are digits, their sum can be at most 18. Therefore, k must be 2, which gives:
y + z = 9
We also know that n is divisible by 4, so the last two digits of n must be divisible by 4. This means that the number formed by yz must be divisible by 4. The possible pairs of digits that satisfy this condition and y + z = 9 are:
45, 81
Out of these, only 45 satisfies n = 23400 + 10y + z. Therefore, we have:
n = 23400 + 45
n = 23445
Hence, the correct answer is option A (23400).
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Community Answer
n = 234yznis a positive integer whose tens and units digits are y and ...
23400/4=5850
23400/5=4680
23400/9=2600
23400/5=4680
23400/9=2600
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