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Soru

Zorluk: ZorHCF and LCM

A municipal election committee is distributing ballots to various polling stations. When they pack the ballots in bundles of 4545, 5454, or 7272, they find that they are always left with 3838, 4747, and 6565 unbundled ballots, respectively. If the total number of ballots printed is the largest possible 4-digit number satisfying these conditions, what is the exact total number of ballots?

Cevap: 9713 ballots

Cevap

9713
The problem describes a scenario where the difference between each divisor (4545, 5454, 7272) and its respective remainder (3838, 4747, 6565) is exactly 77. This means that if 77 more ballots were added, the total would be perfectly divisible by all three numbers. Therefore, the required total is exactly 77 less than a common multiple of these divisors. The LCM of 4545, 5454, and 7272 is 10801080. The largest 44-digit multiple of 10801080 is 97209720 (1080×91080 \times 9). Subtracting the constant difference of 77 from 97209720 gives the final answer of 97139713.

Adım Adım Çözüm

1
Calculate the difference between each bundle size and its corresponding remainder.
45−38=745 - 38 = 7, 54−47=754 - 47 = 7, and 72−65=772 - 65 = 7.
To identify if there is a constant difference, which allows the use of the LCM minus constant method.
2
Determine the Least Common Multiple (LCM) of the bundle sizes 4545, 5454, and 7272.
LCM(45,54,72)=1080\text{LCM}(45, 54, 72) = 1080.
The LCM represents the smallest bundle size that perfectly divides by all three numbers. Prime factorizations: 45=32×545 = 3^2 \times 5, 54=2×3354 = 2 \times 3^3, 72=23×3272 = 2^3 \times 3^2. LCM =23×33×5=1080= 2^3 \times 3^3 \times 5 = 1080.
3
Find the largest 4-digit multiple of the LCM.
1080×9=97201080 \times 9 = 9720.
The problem asks for the largest 4-digit number. Dividing 99999999 by 10801080 yields 9.258...9.258..., so the largest integer multiplier is 99.
4
Subtract the constant difference from this largest multiple.
9720−7=97139720 - 7 = 9713.
Since each division left a remainder that was 77 short of a full bundle, subtracting 77 from a perfect multiple satisfies all three remainder conditions.

Anahtar Kavram

Solving simultaneous remainder problems where the difference between divisors and remainders is constant, by utilizing the LCM and scaling to a specific boundary range.
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Konular

HCF and LCM

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