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09 Nov 2021, 04:28
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What is the value of a positive integer \(x\)?
(1) \(x\) is three times the sum of its digits
(2) \(x\) is a two-digit number
(1) \(x\) is three times the sum of its digits
(2) \(x\) is a two-digit number
09 Nov 2021, 04:28
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Official Solution:
What is the value of a positive integer \(x\)?
(1) \(x\) is three times the sum of its digits
First let's check how many digit number \(x\) can be.
Can \(x\) be a single-digit number? No. Only 0 satisfices \(x=3x\) but we are told that \(x\) is positive.
Can \(x\) be a three-digit number? No. The maximum sum of the digits of a three-digit number is \(9+9+9=27\) and thrice of that is not a three-digit number. The same for four or more digit numbers: thrice the sum of the digits will always be less than the number itself.
Can \(x\) be a two-digit number? Say \(x=10a+b\), then we'd have that \(10a+b=3(a + b)\):
\(\frac{a}{b}=\frac{2}{7}\). Since both \(a\) and \(b\) are positive digits (1, 2, 3, 4, 5, 6, 7, 8, 9), then \(a=2\) and \(b=7\). So, \(x=27\). Sufficient.
(2) \(x\) is a two-digit number. Clearly insufficient.
Answer: A
What is the value of a positive integer \(x\)?
(1) \(x\) is three times the sum of its digits
First let's check how many digit number \(x\) can be.
Can \(x\) be a single-digit number? No. Only 0 satisfices \(x=3x\) but we are told that \(x\) is positive.
Can \(x\) be a three-digit number? No. The maximum sum of the digits of a three-digit number is \(9+9+9=27\) and thrice of that is not a three-digit number. The same for four or more digit numbers: thrice the sum of the digits will always be less than the number itself.
Can \(x\) be a two-digit number? Say \(x=10a+b\), then we'd have that \(10a+b=3(a + b)\):
\(\frac{a}{b}=\frac{2}{7}\). Since both \(a\) and \(b\) are positive digits (1, 2, 3, 4, 5, 6, 7, 8, 9), then \(a=2\) and \(b=7\). So, \(x=27\). Sufficient.
(2) \(x\) is a two-digit number. Clearly insufficient.
Answer: A
