Netherlands
The letters a,ba,b and cc stand for non-zero digits. The integer abcabc is a multiple of 33 the integer cbabccbabc is a multiple of 15,15, and the integer abcbaabcba is a multiple of 8.8. What is the value of the integer cba?cba?
Answer:
576576
- We know that a number is divisible by 88 if it's last 33 digits are divisible by 8.8.
Given, abcbaabcba is a multiple of 8.8.
Therefore cbacba is a multiple of 8.8. - Also, abcabc is given to be a multiple of 3.3.
Since the sum of the digits of abcabc and cbacba are the same, cbacba is also a multiple of 3.3.
Therefore, cbacba is a multiple of 24.24. - We are given that cbabccbabc is a multiple of 1515 and c≠0c≠0 (given).
⟹c=5⟹c=5
Now, cbabccbabc is a multiple of 1515 therefore cbabccbabc is a multiple of 3.3.
⟹⟹ sum of digits of cbabccbabc is a multiple of 3.3.
Also, a+b+ca+b+c is a multiple of 3,3, therefore, c+bc+b is a multiple of 3.3. - The three-digit multiples of 2424 starting with 5,5, which are the possible values of cbacba are 504,528,552,504,528,552, and 576.576.
Out of the above possible values of cba,cba, only 576576 has c+bc+b as a multiple of 3.3. - Hence, the value of the integer cbacba is 576.576.
