Identical digit multiplication
77 is multiplied by another two-digit number with identical digits. What is the product?
77 is multiplied by another two-digit number with identical digits.
The third digit of the product, counting from left to right, is a 3.
What is the product?
This problem is adapted from the World Mathematics Championships
Trying out numbers
77$\times$11 = 770 + 77 = 847, which does not have a third digit of 3
77$\times$22 = 847 + 847 = 1694, which does not have a third digit of 3
77$\times$33 = 1694 + 847 = 2541, which does not have a third digit of 3
77$\times$44 = 2541 + 847 = 3388, which does not have a third digit of 3
77$\times$55 = 3388 + 847 = 4235, which does have a third digit of 3
So the product was 4235.
Multiplying 77 by a number written $kk$
Writing out the mutliplication like this,
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All of the mutliplications will be $7\times k$, plus carried digits and the $0$ as shown below. The third digit in the product will come from the tens digit of the pink box below added to the units digts in the green box.
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So the last digit of the number which is the tens digit of $7k$ plus twice the units digit of $7k$ is a $3$.
If $k=1$, then $7k=7$, and the tens digit plus twice the units digit is $0+14=14$ which has last digit 4.
If $k=2$, then $7k=14$, and the tens digit plus twice the units digit is $1+8=9$ which has last digit 9.
If $k=3$, then $7k=21$, and the tens digit plus twice the units digit is $2+2=4$ which has last digit 4.
If $k=4$, then $7k=28$, and the tens digit plus twice the units digit is $2+16=18$ which has last digit 8.
If $k=5$, then $7k=35$, and the tens digit plus twice the units digit is $3+10=13$ which has last digit 3.
So $k=5$, and the product is $4235$.
Using the 11 times table
Two-digit numbers with identical digits are precisely the multiples of 11. So 77 has been multiplied by 11$n$ for some value of $n$.
77$\times$11$n$ = (77$\times$11)$n$ = 847$n$. So 847$n$ = _ _ 3 _ .
The 800 will not contribute to this digit, so we can consider the 47 times table.
47$\times$2 = 94
47$\times$3 = 141
47$\times$4 = 188
47$\times$5 = 235
So $n$ = 5, and the product is 847$\times$5 = 4235.