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Zorluk: OrtaClassification of Numbers

Consider all positive two-digit integers where the sum of their digits is exactly 1010. How many of these integers are prime numbers?

Cevap: 3

Cevap

3
By listing all two-digit numbers with a digit sum of 10 (19, 28, 37, 46, 55, 64, 73, 82, 91) and testing them for primality, we find that only 19, 37, and 73 are prime. The number 91 is composite (7×137 \times 13), resulting in exactly 3 prime numbers meeting the condition.

Adım Adım Çözüm

1
Identify all two-digit positive integers whose digits sum to 10.
The numbers are 19, 28, 37, 46, 55, 64, 73, 82, and 91.
Systematically listing the pairs of digits (from 1 to 9) that add up to 10 ensures no valid numbers are missed.
2
Evaluate each number to determine if it is prime or composite.
Even numbers (28, 46, 64, 82) and multiples of 5 (55) are composite. The remaining odd numbers to check are 19, 37, 73, and 91.
Applying basic divisibility rules for 2 and 5 quickly eliminates obvious composite numbers.
3
Test the remaining odd numbers for primality.
19, 37, and 73 are prime numbers. 91 is a composite number because 91=7×1391 = 7 \times 13.
A prime number has exactly two distinct positive divisors: 1 and itself. 91 is a common trap as its divisibility by 7 is not always immediately obvious.
4
Count the total number of prime numbers identified.
There are exactly 3 prime numbers in the set: 19, 37, and 73.
This directly answers the specific question asked in the stem.

Anahtar Kavram

Identification of Prime and Composite Numbers
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Konular

Classification of Numbers

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