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Two and Three-digit numbers | AIME I, 1997 | Question 3

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1997 based on Two and Three-digit numbers.

Two and Three-digit numbers - AIME I, 1997


Sarah intended to multiply a two digit number and a three digit number, but she left out the multiplication sign and simply placed the two digit number to the left of the three digit number, thereby forming a five digit number. This number is exactly nine times the product Sarah should have obtained, find the sum of the two digit number and the three digit number.

  • is 107
  • is 126
  • is 840
  • cannot be determined from the given information

Key Concepts


Twodigit Number

Threedigit Number

Factors

Check the Answer


Answer: is 126.

AIME I, 1997, Question 3

Elementary Number Theory by David Burton

Try with Hints


First hint

Let p be a two digit number and q be a three digit number

here 1000p+q=9pq

\(\Rightarrow 9pq-1000p-q=0\)

Second Hint

\((9p-1)(q-\frac{1000}{9})\)=\(\frac{1000}{9}\)

\(\Rightarrow(9p-1)(9q-1000)\)=1000

Final Step

from factors of 1000 gives 9p-1=125

\(\Rightarrow p=14,q=112\)

\(\Rightarrow 112+14=126\).

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